@Ryan: also, what would be the consequences of having complete problems for $NP\cap coNP$?

@Ryan: OK, I understand. Then, what do we need to proof collapsing in the polynomial hierarchy in general?

in the draft version, chapter 5, the polynomial hierarchy and alternations, page 5.2(92), where it says "Note that $\sum_2^p contains both the classes NP and coNP". There is no proof, but this implies that a complete problem collapses the hierarchy to the second level.

Thanks for the info. But what I wanted to point out is that for the "lower" classical complexity classes (PSPACE and below) this is the best, is that correct? Although the NQP=coC_{=}P result seems to be at a really low level.

Well, I have a bias here to be honest but I propose this as a result for the 2005-2010 period. First, to my knowledge, this is the best relation we have between classical and quantum classes. There are other good results on upper bounds for BQP, but this is the only result where a quantum complexity class is completely characterized. Second, although I don't know the complete details, the proof seems to be non-relativizing. And that's important because we can try to learn from here and use it to proof other non-relativizing results. Although, other people already tried that.

Ryan, yes your right, it's not clear. I've edited the post. Also, thanks for the nice example on CAPP.

Peter, thanks for the answer. But in general, can we ask something like: Let A and B be a complexity classes of promise problems such that $B\subseteq A$, but not known to be strictly included. If we assume $A\neq B$ then there exists a promise language $L$ which is intermediate for $A$? In other words, is there a theorem like Ladner's but for any complexity class (including randomized and quantum) considering promise languages?

Yes, it actually should read "there exists" instead of "for any".

Yes, that's true. It is a very strong thig to say. I'll edit that part. Thanks