Abilities demonstrated by magical girls fall broadly into two different classes -- material magic, which involves the materialization of magical implements or the stylized transfiguration and reinforcement of existing objects, frequently employing the Soul Gem as a medium for the magical effect; and support magic, an umbrella category under which all other abilities are placed, influenced generally by the parameters of a magical girl's wish. Please note that this distinction is arbitrary and fan-made; canonically, no such categorization is applied.
This one allows to create or transform objects using a gem's power. We have seen many applications of this magic already - the magical girl costumes, all the weapons they create and possibly Kyoko's food as well. Sayaka's bat and the binoculars Kyoko used are examples of a enchantment/transformation of a mundane item.
This one was a specialty of Mami's. She enchants Sayaka's bat so that it automatically protects her, and can be planted in the ground for a more permanent barrier. Energy ribbons are used twice - first to catch the suicidal lady and the second time to bind Homura to prevent her getting in Mami's way. She uses plantlike tendrils to bind Gertrud and lift Charlotte. Her ribbons with a chain-motif and a lock bind Homura and presumably seal her powers. Also, Mami's last "Tiro Finale" appeared to be a binding variant of the direct damage move she used previously. Mami's proficiency in this area leaves a question - what kind of wish could affect it so much? "I wish I fastened my seatbelt" is hardly a correct answer. Since she wishes for her life to be saved, the explanation can only be derived indirectly- hence, ribbons that bind bring to mind 'strings of life' (she didn't want her string of life to be cut or she tried to bind herself with life). The same goes with her ability of creating barriers, spaces where life is protected. Others have speculated that Mami's wish has its origins in Buddhism - if one wishes to 'continue to live,' it means that they are attached to life. Mami's ribbon powers thusly reflect her desire to connect to life.
This one consists of few skills every girl can use, but the power of each skill heavily depends on their wish.
We see Mami and Sayaka using it. Mami used it outside battle and took some time to heal minor wounds on Kyubey, while Sayaka used it in battle to fully recover from major damage Kyoko inflicted. Sayaka's power is explained by Kyubey - her wish gave her very strong regenerative abilities. It is currently unknown if Mami's ability was a result of her wish or not, however Mami implies her wish saved her life after being injured in an accident, meaning her wish was also tied to regeneration and healing.
Kyubey himself claimed that it was possible for every magical girl to fight even if every drop of blood was drained from their body due to the fact their bodies are just meat puppets they can repair with their magic. Likewise, the feeling of pain is dulled considerably to allow them to continue fighting through otherwise debilitating injuries, and can even be switched off completely. But as Kyubey states, that results in their movements becoming more sluggish. Sayaka is the only one that has been seen to actually switch off their perception of pain.
Homura is confirmed to be able to either move faster than time or stop time. She can bring people into the freeze with her if she is touching them (possibly this only works with hand contact, as demonstrated by the fact she was able to move Sayaka away from Kyoko's attack without Sayaka realizing what happened, meaning Homura would have dragged her by the clothes or some other method. More likely however is anything she's in direct contact with when she stops time initially will be in Homura's paused time frame reference. If they then lose contact, then they return to the normal time frame, and things can only be brought into Homura's time frame when she pauses time initially. This would explain why she could move Sayaka without her being aware of it, be hesitant about pausing time when Kyoko had grabbed her, then later warn Kyoko not to let go while they were escaping from Oktavia).
This type of magic is granted to her by the fact her wish was related to time.
Due to the way stopping time works, it can give the impression Homura is able to teleport/blink around, when in fact she's just moving while time is stopped.
It is likely that Homura is surrounded by a "bubble" that time stoppage does not apply to. This would explain why normal physics, talking, breathing, and exempting people from stoppage through direct contact work when time is stopped, as well as the behavior of objects such as bullets and rockets that Homura fires while time is stopped. The bubble would not allow distant photons to travel towards Homura, so it is possible that Homura uses magic as radar to detect objects, and then summons photons (in the same way a magical girl would summon guns) to display detected objects to herself and anyone she brought into the bubble.
Time stop/Time bubble (unconfirmed - almost definitely wrong)
Seen twice in ep.5. First, Sayaka uses it for few seconds to stop Kyoko's spear with a sword (that's first time we see water drops hanging in the air). Then Homura uses greatly advanced version of it - she leaves message to Madoka and then saves Sayaka, also giving her some time to rest (note how Sayaka is laying when Kyoko is about to strike, and how she's resting behind after Kyoko lands). Although full time stop does leave some major inconsistencies (Homura could use it while hunting Kyubey and escaping Mami's bind), Time bubble (where time keeps flowing not personally for caster, but in some area of simple form, which can't be left without resuming time) explains most things. Is Homura is really proficient in this, it makes her wish heavily related to time, which supports the time loop theory.
Demonstrated by Sayaka and Mami, and Kyoko a little bit as well. Magical girls seem to be able to produce an unlimited amount of their favorite weapons (rifles for Mami, swords for Sayaka, spears for Kyoko). They can produce them directly from their Soul Gem (as demonstrated by Kyoko) or from their clothes/hammerspace, as demonstrated by the others. They can also modify these weapons, such as making a large pistol for Mami or gigantic spears for Kyoko.
A sub-application of this power may be hammerspace, such as Homura storing firearms in her shield.
Another sub-application would appear to be the ability to imbue items with power, such as when Mami turned a regular baseball bat into a more regal looking mace of solid steel that could sprout spikes.
So far we've seen that both Kyoko and Mami are able to generate barriers/fences to protect people (usually Madoka) or to keep them out of battles. Mami particularily seemed to specialize in this, able to turn her ribbons into intricate shapes and ropes or to produce strong ones around targets. Homura seems to have the ability to make magical shields to block attacks as well. In Oriko Magica Homura is seen displaying the same power as Mami, creating a barrier around Madoka.
Kyubey has used telepathic communication and acted as a relay allowing others to use it. Kyoko has also used telepathy to communicate with Madoka when Kyubey was not present. Madoka did not reply telepathically at that time, although she did so earlier when Kyubey was relaying.
It is likely that magical girls can send messages, normal humans can only receive them unless they are assisted by a relay, and Kyubey has technology that gives him a wide range of telepathic abilities. It is unclear whether Kyubey's abilities include outright mind reading on normal humans (Kyubey definitely can't read magical girl Homura's mind), or whether they are purely technological or powered by using technology to harness magic.
Alternate vision of Homura's skill. Time manipulation version does not explain Homura's appearing out of nowhere after being hit by a building in the prologue as well as some of her movements during the Charlotte fight. She might have both abilities, which leads to even more powerful wish and may have been foreshadowed by Sayaka's quip in Ep.1. Re. Homura and Madoka: "You're friends, meeting each other again after having traveled through space and time!"
Spatial Reconstruction (unconfirmed)
Referred to as Spatial Reconstruction Magic (空間再構成魔法 Kūkan Sai-kōsei Mahō) by Asian fans. In plain language, a user of this class of magic temporarily becomes the 'goddess' of a limited domain of space, gaining the ability to change, create, or destroy objects within it -- including distances and spatial coordinates. Humans appear to be shielded from manipulation to a degree, though their position may still be freely rearranged. Distinct from the 'object modification' described in the Material Magic section above -- where the qualities of an existing tool or implement are reinforced as it temporarily undergoes a highly stylized transformation -- inanimates affected by Spatial Reconstruction experience a permanent, unstylized alteration.
Speculah regarding Spatial Reconstruction began to surface after Homura intervened in the confrontation between Sayaka and Kyoko in Episode 5: Viewer discussion concluded that the complete redrawing of the wall layout, the floor, and the lighting in the background art following Homura's use of 'teleportation' was suspiciously unnecessary -- it would have been far easier for the animation staff to reuse previously completed artwork. If this was not in fact a production error, the theory followed that the ability used by Homura to save Sayaka may have been something on the order of reconstructing the space within the corridor, rather than mere teleportation or the use of a 'Time Bubble.' It should be noted that the alteration of the wall is presented unchanged in the scene recap at the beginning of Episode 6.
Outside the context of theory, Witches and their Familiars canonically employ magic within the spectrum of Spatial Reconstruction in the manifestation of their Labyrinths. Movement subjectively experienced by persons drawn into a Witch's Labyrinth does not necessarily indicate or overlap any analogous travel within 'real' space; observed space within a Witch's domain comprises thus of 'spoofed coordinates.' Ergo, the manipulation of the position or lay of objects in a given space figures fundamentally into the magical repertoire of a typical Witch. The 'cartoonization' attack used by H.N. Elly and her Familiars against Madoka may classify an offense implementation of Spatial Reconstruction.
Mind Trick/Illusion (Confirmed)
In other terms "Jedi Mind Trick" or the mental ability to influence and manipulate the mind of others to believe you. Kyoko's mental ability has been described as being the power of persuasion and hallucination. Perhaps even the ability to persuade others to follow orders according to one's will without forcing them. It is also believed that she can shine light into illusions or lies, to unveil the truth or what is hidden.
According to Kyoko her father had this power because of her wish. We do not know the extension of his powers. Our only clue is that whoever listened to Kyoko's father they would listen and believe him (no matter how absurd his statement or lofty his ideals). We dont know the duration of the spell, its effects on the individual, or if there is any mental damage (and if there is, if it is permanent on the target).
As a result of the nature of Kyoko's wish she inherited a similar mental ability as her father (a hidden ability). Her secondary ability has been described as the ability to charm others, to persuade them to see her point of view, rather than directly forcing them to follow her will. Kyoko's subconscious suppression of this secondary ability is a consequence of the traumatic events of watching her father kill her family and then hang himself after learning of his own mental powers. Kyoko was aware of her secondary ability but she lacked proper training to use proficiently until she met Mami.
Kyoko's secondary ability was never shown or told in the anime, however, a magazine article corroborated the possibility that at one point in her life Kyoko did indeed use such powers. Then when the Drama CD "Farewell Story" came out, Kyoko displayed her secondary abilities to her full power. Her illusory powers includes the ability to confuse her opponents and able to split herself into multiple images to confuse her enemies.
Kyubey calls this power "enchantment" in Puella Magi Madoka Magica: The Different Story and the manga goes into detail as to why Kyoko lost her magical power. Kyubey explains to Kyouko that the reason she lost her power, is because when she consciously rejected her own wish, her subconscious mind also suppressed her own magical ability. It is speculated that as long as Kyouko continues to regret her wish she wont be able to use her magic as she did before. With the traumatic event on her life, it is unlikely that she will stop regretting making her wish.
The extensive use of time travel in the show creates some paradoxes. To resolve them, it is necessary to use some sort of "meta time" so that the causes of events are not undone by the events themselves.
One speculated solution is dividing time into chronological time and perceived time. Chronological time is time that happens within the time-space continuity of the universe the show takes place in. Perceived time is the way characters or viewers experience events. Anyone who transcends timelines has personal perceived time, including Homura, Madoka, and the audience. Time travel involves travelling through chronological time and between chronological timelines, but not through perceived time, which continues to progress linearly for each individual.
Under this system, the events that turned Madoka into a god actually happened, and are part of Madoka's and Homura's perceived time continuities, but are no longer part of the universe's time-space continuity after they happen.
Travel to Parallel Universes
There are some bizarre elements that havent been properly addressed in the anime series. In one case is the issue that the production notes state that in one timeline Kyousuke was a guitarist. It was never addressed in the Anime but in the alternate-Oktavia's witch card does mention a guitar, and we have no reason to assume it refers to anyone but Kyousuke. On this reason Kyousuke as a guitarist is considered to be canon. Also, the settings of Oriko Magica are slightly different, featuring new Magical Girls and Witches. One possible explanation for this is that Homura is not only going back in time but also going to parallel universes (different universes) contained within the multiverse. In episode 11, Kyubey refers to the creation of parallel universes when explaining the results of Homura's time travel.
Gen Urobuchi stated that he didn't really think too much about the mechanics of Homura's time loops, but does refer to them as parallel worlds. Chiwa Saito also refers to them as parallel worlds in the BD 6 audio commentary.
Schrodinger's Cat is a thought experiment used as an example to explain the possible existence of parallel universes. This video tries to explain the complexities of Newtonian Laws and Quantum Mechanics.
The Madoverse, The Homuverse, and the Multiverse
It has been speculated that the events in the anime series are taking place in parallel universes, that Homura is not time traveling as so much as jumping to other universes similar to the previous ones, but with slight differences. However, what if they are not just parallel universes? What if there are other universes existing simultaneously, with their own set of laws and rules without cancelling each other? By the end of the series the entity known as Ultimate Madoka is said to have rewritten the Old Universe and to have created a New Universe with a new set of laws (Law of Cycles). But what if Madoka didnt rewrite the universe and instead created a new one? If so, what happened to the original universes? And not only that but what happened to the previous parallel universes? Did they collapse because of paradoxes?
Perhaps the parallel universes didn't cease to exist just because Homura abandoned them, so perhaps the Old Universe (where witches do exist and there is no hope) continues to exist alongside the New Universe, that is the Madoverse, but without able to interact with or able to cancel each other out. The difference from the old speculation is that Homura and Madoka from the old universe are now inhabiting a New Universe, the Madokaverse, speculating and implying that the Old Universe and the New Universe are continuing their separate existence from each other with different rules/laws and with different memories and some similar histories. They simultaneously exist, yet they work independently from each other.
But that is not the end of it, in the new movie "The Rebellion Story", the entity known as Homura also "rewrites" the universe (or in this case, creates a New New universe, the Homuverse), thus expanding the known plethora of universes within the franchise. Homura's new universe is similar to the Old Universe and the Madoverse, but this one is Homura's ideal version of it. If we assume that the previous universes did not cease to exist and that they continue to exist independently, then it is possible that the Old Universe, the Madoverse, and the Homuverse are now inhabiting with other universes (and parallel worlds) continuously without interruption. They are part of the Multiverse. And as a result this Multiverse contains many different versions of other universes, some of them similar, some of them completely different, and some of them governed by a different set of natural laws that would be completely bizarre to our own.
It is highly speculated that if a Puella Magi's body is destroyed or heavily damaged, but their Soul Gem remains not only intact but untainted, a new body could be created to replace the lost one with magic. According with Kyubey their bodies are soulless empty shells that are only controlled by the Soul Gem, where their souls reside. If their bodies are then puppets, then it is possible to suspect that they could be replaced with new ones so Kyubey can continue with his quota to collect energy, an untainted Soul Gem with no body would be unprofitable for Kyubey's plan to collect more energy.
Gen Urobochi has said in a tweet that if a magical girl's body was completely destroyed, her consciousness could reside in the Soul Gem and use magic to emulate the body's senses. However he said eventually it would have been "unable to do so and lose awareness". It is vague wether this is referring to Witching or simply just dissipating. Sayaka's comatose state from when her Soul Gem was seperated from her body probably means that this ability must be learned.
Kazumi Manga: Immortality
It is theorized in Puella Magi Kazumi Magica that if a girl's body was preserved, any damages or wounds healed and repaired, and her Soul Gem still remained intact, then technically she would be immortal.
Kanna Hijiri: Artificial Lifeform
In Puella Magi Kazumi Magica, it is revealed that Kanna Hijiri is the product of magic/wish. She is a duplicate of Nico Kanna but with a different set of memories. A very interesting question arises from this issue. If magic or a wish can create a new body (or an artificial one indistinguishable from the original), then it is possible it can also create a new soul (an artificial soul, a product of magic). The artificial body and artificial soul is not just indistinguishable from real ones but it does not limit the artificial person from contracting with Kyubey. The implications are clear, Kyubey's race have the knowledge and the power to create new life, if so, why not use the same power to reverse entropy without sacrificing real human beings? For the moment we have to assume that this issue was already visited by Kyubey's race and was considered insufficient to solve the entropy crisis for some reason.
The only clue we have to explain this phenomena is what Juubey states in Chapter 18 of Kazumi Magica, "As long as you possess a human heart, you can make a contract. Even if you aren't a human. But the results are the same as the others. One point or another, you'll become a witch and die.". One interpretation is that theoretically, any being capable of producing the same type of emotions as humans would be able to contract, fall into despair, and produce the same type of energy to counter-act entropy.
Kyubey's race cannot comprehend or reproduce human emotions, if they did they wouldn't have bothered to stay on earth in the first place. It's fair to assume any artificial human created by the Incubators alone would lack a proper human soul and display no magic potential even if the genetic make up was identical to that of a real human. Kanna Hijiri on the other hand, was born from the wish of an exhisting girl with emotions and whatever is working behind magic just made a copy of Nico's soul with a different set of memories.
What makes this really scary is that Kazumi being an artifical girl made with magic and witch parts has a soul of her own, this could mean that magic and souls are actually different forms of the same element. This would explain the corruption, the girls' souls are actually converting into magic thus leaving their darkest and heaviest emotions (the evil in their souls) assuming full control and transforming them into witches.