Quantum computers, parallel computing and exponential time
Why can't a quantum computer solve exponential problems in sub-exponential time? Factoring is believed (by some) to take classical time $2^{n^\epsilon}$ for some $\epsilon &gt; 0$, while Shor's algorithm takes time $O(n^3)$. Does that count?

Is it at least theoretically possible to imagine a computer able to solve EXPTIME-complete problems in constant time? Sure, consider a Turing machine with oracle access to all EXPTIME problems (i.e. the oracle accepts the code of an EXPTIME machine and an input, and returns the output). This doesn't lead to any contradictions. Is this realizable in practice? Probably not.

Can quantum computers do parallel computing? It is known that BQP$\subseteq$PP$\subseteq$EXPTIME, and most people expect these inclusions to be proper. Under this conjecture, there are problems solvable in classical exponential time but not quantum polynomial time. In particular, EXPTIME-complete problems shouldn't be in BQP.