Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

It is known (Boppana et a;., 1987) that if every language in coNP has a constant-round interactive proof system, then the polynomial hierarchy collapses. On the other hand, Lund et al. (1992) have shown that #SAT, the #P-complete function that outputs the number of satisfying assignments of a Boolean formula, can be computed by a linear-round interactive protocol. As a consequence, the coNP-complete set SAT has a proof system with linear rounds of interaction. We show that if every set in coNP has a polylogarithmic-round interactive protocol then the exponential hierarchy collapses to the third level. In order to prove this, we obtain an exponential version of Yap's result (1983), and improve upon an exponential version of the Karp-Lipton theorem (1980), obtained first by Buhrman and Homer (1992).