As I said above, Deutsch believes that we can make computers which use quantum effects to outdo conventional computers - to compute Turing-computable things faster than any Turing machine could. But he does not believe the brain does so. So in Deutsch's view, the brain is no more powerful than a Turing-equivalent computer.
Penrose, on the contrary, believes it is. He also believes that we could build a Deutsch-style quantum computer - Emperor's New Mind p 401. But he doesn't believe that its kind of quantum effects would be any use to the brain - Emperor's New Mind p 402.
What effects does he want to use? In last week's lecture, I said that one of the problems no Turing-equivalent computer can solve is this: Consider a set of polygons. Is there an algorithm that will tell you whether how to given any set of polygons, it's possible to cover an infinite plane with no gaps and no overlaps? Obviously one can do this for some sets, e.g. that containing just one square. But it can't be done in general.
Going on from that, it may be that for some particularly wierd sets of polygons, there are no rules that tell you how to cover the whole plane with them. I.e. this problem would also be non-Turing-computable. Penrose speculates that may be the case for the Penrose tiles shown on pages 136-137 of Emperor's New Mind. He makes this speculation on page 438 (bottom paragraph). Apparently, when laying down the tiles, you may have to look arbitrarily far ahead to check there are no overlaps or gaps.
Suppose now that we could find a crystal which was built up from unit cells with the same shape as Penrose tiles. If we believe no Turing-equivalent computer could work out how to cover the plane with Penrose tiles, then, just by being able to crystallise itself, this crystal is doing something which is not Turing-computable. Perhaps it uses some kind of holistic non-local quantum effect to do so (page 437).
In which case, perhaps the brain uses the same kind of non-local quantum effect when it does intuitive, judgemental thought (see pages 411-413 for examples).
There are a lot of assumptions here!