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).