“If you think you understand quantum theory, you don’t understand quantum theory”, words attributed to Richard Feynman and as true now as when they were uttered. The quantum world is replete with examples that make a mockery of our intuitive understanding of the world. From uncertainty principles saying you can’t know exactly where a particle is and how fast it is going to electrons that can pass through solid barriers. Light that is both a particle and a wave, virtual particles and quantum foam, it’s a crazy world down there. Unfortunately this very impenetrability makes quantum theory ripe for misrepresentation and co-option by those who use it as a smokescreen to cover the initial implausibility of their claims.
The mysterious nature of the quantum realm makes it easy to think that almost anything is possible and so any claim that attaches to itself the concept of quantum mechanics becomes more reasonable by association. Telepathy, clairvoyance, precognition, paranormal claims of all kinds have been subject to attempts to provide them with a plausible basis by tying them with quantum strangeness. Unfortunately, most of the unusual properties shown by atoms and subatomic particles are incredibly unstable and have a tendency to collapse on themselves and cancel out any spooky behaviour in a process known as decoherence. Mere interaction with other particles can provide an excuse for this process, unless you go to extraordinary lengths in a purpose built laboratory, particles tend to collide.
Another barrier to the likelihood of quantum mechanics as a framework for the supernatural is the De Broglie Wavelength, this is the effective wavelength of any body and is related to the wave/particle duality of matter and has a direct impact of the distance scales over which quantum effects are observed. For very small objects, like electrons the De Broglie Wavelength is large and quantum effects are pronounced, as objects increase in size this wavelength rapidly decreases until for classically sized objects, like brain cells, it essentially vanishes, taking all quantum weirdness with it. That’s a pity, I’d love to be able to walk through walls, oh wait, we have doors for that.