Quantum fanfiction is storytelling in which the theory of quantum mechanics is both mentioned, perhaps even explained a bit with questionable success, and made directly relevant to the story's plot. In otherwords, the word "quantum" isn't merely a throwaway reference inserted to prove that whoever said it was smart, a technique seen in fanfiction of all sorts. All quantum stories are of necessity science fiction or science fantasy. It is not necessary for the way in which quantum mechanics is used in the story to be at all accurate, but it is important that the story present a significantly bizarre alteration of commonly accepted reality and attempt to link it to an aspect of quantum theory.
The significance of such a thing as quantum fanfiction lies in what it reveals about the fandom that embraces it. It could be fairly said that those who enjoy quantum fanfiction readily grasp abstract concepts and are able to consider and appreciate unusual views of reality (as in science fiction), plus share a taste for high-order intellectual amusements. Or at least that is TAG's take on the matter.
DC Comics' superhero Doctor Manhattan, from the 1986-1987 Watchmen graphic novel, was the first fictional character for whom quantum theory was inherently relevant. It is possible that this character and the later TV series Quantum Leap were the genesis of quantum fanfiction.
Jokes about quantum mechanics appear in "Valley of Love and Delight," by Richard Lobinske, and "Hells Angels?" by Thomas (though the joke is incomplete in the latter: How does a quantum physicist catch a lion? He sets up the cage and waits, because at any given moment there is a non-zero probability that a lion will be in the cage.)