Are provable quantum speed-ups possible for classes larger than NP?
For total functions it is impossible to achieve more than a polynomial separation (i.e. $D(f) = O(Q^6(f))$) in the quantum query model, for more info see this cstheory blog post. With promise problems, you can have exponential seperations between the quantum and classical query complexity. However, classical query complexity is sometimes very deceptive in its correspondence to complexity classes; as are partial functions in general. If you don't allow promise problems, you can't get exponential separations and your relevance to the big classes is questionable. You can never expect more than an exponential seperation, because a quantum algorithm can be simulated by a classical computer with exponential overhead (actually, in PSPACE so looking above that is dubious).