Okay, so, let me try to enumerate these.

(1) The hardest part of quantum mechanics to understand is intimately tied to our nature as subjects. Quantum mechanics is an entirely deterministic theory about waves in superpositions which even explains how, why, and when it disappears and averages out into classical probabilities. Except it doesn't tell you why we don't see your television as 50% in your bedroom and 50% in your living room. For that we start to invent things like many universes, half of which have a TV in your bedroom and half of which have a TV in your living room -- or Penrose's idea that perhaps some magical interaction not yet observed forces something as large as a TV to "make up its mind" due to its gravitational warpings. It's just the fact that your subjective experience doesn't see this TV as half in one place, and half in another, where the wavefunction description fails. The rest is clean, deterministic, and precise.

(2) The second hardest part of consciousness to understand is something we basically take for granted in the hardest part of quantum mechanics. Compatibilist theories of free will make you feel warm and fuzzy but they don't really feel like they address the fundamental mystery: How can a deterministic system be said to "choose" something when that's a fuzzy airy-fairy notion and the actual outcome was written down in the book of history 13.7 billion years ago, full stop? Why do we feel like we've got a million options open when in fact this is false, false, utterly false, moronically-out-of-touch-with-reality crazy false? But in quantum mechanics, the idea that a quantum system has many options open, and can even "sniff around" over the whole space, and that it might make a difference if it does or not -- these things are very acceptable even if you take a deterministic approach to wavefunction collapse. You could maybe in principle go from "warm and fuzzy but I don't get it" to "weird and fuzzy but I think I get it," if the brain is intrinsically quantum.

(3) The hardest problem of consciousness -- how we reconcile our touchy-feely alive experience of red with a machine's cold unemotional ones and zeros -- is very easy to solve by saying "we aren't a machine; subjectivity is something more than a program." In fact, since programs can carry out programs, if consciousness is mechanical we might well ask, how do you know that your brain isn't a different conscious process which happens to be carrying out your own conscious process, and might someday soon get bored and kill you out of its own boredom and desire to live authentically? (This absurdity is also at the heart of Searle's Chinese Rooom argument.) The problem is that machines are such a good model that we have no idea what to replace them with. If quantum systems could really feel in a way that machines really can't, this would solve the problem neatly.

(4) One minor problem in consciousness is about how I'm able to hear things as I am seeing them -- multithreading would suggest that we have multiple consciousnesses doing the separate tasks separately; multitasking would suggest that we switch between these things so fast that it gives an illusion that they all come together, but I can't really do both at once. Quantum mechanics offers a tantalizing hint; for example, in a superconductor, two electrons of opposite momentum "get together" to form a larger particle called a Cooper pair. (You're advised not to think too hard about how they "stay together" with opposite momenta, because It's All Just An Excitation Anyways.) That larger particle can even do something which electrons never can -- it can form a Bose-Einstein condensate: it can conduct current without resistance.

(5) Everybody who watches Star Trek wonders what would happen if someone accidentally forgot to dematerialize you before re-materializing you down on the planet -- which one would be the right "you"? Quantum systems cannot be cloned in this way. The closest we can get is to identically prepare a system -- i.e. take another baby with the same genetic code and give it all of the same life experiences, and kill the ones who don't make the same choices in the same order. Without one of those, you can't clone an arbitrary unknown system into two copies without measuring it and thus forcing it into a classical state. If consciousness is quantum, then the classical state is dead. But if consciousness is just a program, then we're faced with a much scarier situation: in the transporter, you will actually die, but that's okay, because we'll create an impostor who will sleep with your wife for you -- in other words, it's okay because we'll never know the difference, and you'll be dead.