It is shown that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time collapses to the second level of the polynomial-time hierarchy, has polynomial-size circuits, and has publishable proofs (EXPTIME=MA). It is also shown that BPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, it is shown that BPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in MA/P. The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs, and classical complexity classes. An important feature of this proof is that it does not relativize.