It is well-known that the set of languages having two-prover interactive proof systems, in which the verifier runs in polynomial-time (MIP), is NEXP. But are there bounds known on the power of such interactive proofs when the provers are restricted in power? E.g., what is the class of languages that admit two-prover interactive proofs with polynomial-time provers?
More precisely, let's say that on an input x I allow the provers arbitrary pre-computation time, but once the interaction with the verifier starts they are restricted to using polynomial space (including storing the results of any pre-computation), and polynomial time to compute their answers to the verifier's question. Let's also assume that these space and time bounds are a fixed polynomial in the length of the questions that will be sent by the verifier (instead of the length of x), in order to preclude a more trivial solution in which the verifier would somehow exhaust the prover's space bound by asking polynomially more questions.
Clearly, this is enough for NP. What about PSPACE? If there was only the space bound they could do it, but what with the time bound? Are there any interesting results in that direction?
I am also interested in other limitations that one might consider on the provers. One of those would be the amount of communication prover->verifier, which I think has been thoroughly studied in the context of PCPs. What are the other interesting constraints?