Is there a version of MIP=NEXP with relatively efficient provers?
Yes. ​ ​ ​ This paper constructs "a one-round succinct MIP of knowledge, where each prover runs in time $t\cdot \operatorname{polylog}(t)$ and space $s\cdot \operatorname{polylog}(t)$ and the verifier runs in time" ​ $\operatorname{length}(m) \cdot \operatorname{polylog}(t)$ .