There are arguments, notably in "The Emperors New Mind", that assert that consciousness must be based on quantum mechanics in some ill-defined way and therefore that "algorithmic" computers can't therefore be conscious.
NB I didn't find this argument particularly strong, but it's a long time since I read it. The brain would have to be doing something spectacularly strange for it to be impossible for us to emulate in one way or another.
Out of context it indeed sounds rather fuzzy and weak. It's not even intuitive to me that consciousness comes first. Maybe they are even independent. I should probably read the book though.
More generally, I don't think invoking quantum effects is an especially strong argument for impossibility in the general case. We already use quantum effects to some extent and make progress on quantum computation. It may be evidence of greater difficulty, but not of impossibility. Unless there is a strong case for some particular quantum effect being unharness-able.
Pretty much everyone I've talked to who knows about AI (including a former Penrose student who was a classmate on my postgrad AI degree) strongly refute the claims in The Emperor's New Mind. I believe Penrose was at Oxford when he wrote it (in 1989), in which case he would surely have come across David Deutsch's universal quantum computer (from 1985).
At the scale an actual brain works, an actual exponential slowdown because of "quantum brains" would make the problem flat-out impossible. An actual human brain has an average of 100 billion neurons. To simulate an actual brain, you'd get an 'additional' slowdown of 2^100,000,000,000. The problem would go from possible in maybe 20-50 years to impossible unless we manage invent some sort of quantum/neural computer.
Not that quantum mechanisms in brains are likely, so probably not an issue.
https://en.wikipedia.org/wiki/The_Emperor's_New_Mind
NB I didn't find this argument particularly strong, but it's a long time since I read it. The brain would have to be doing something spectacularly strange for it to be impossible for us to emulate in one way or another.