Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

If every language in coNP has a constant-round interactive proof system, then the polynomial-time hierarchy collapses [R.B. Boppana, J. Håstad, S. Zachos, Does co-NP have short interactive proofs? Information Processing Letters 25 (2) (1987) 127–132]. On the other hand, the well-known LFKN protocol gives O(n)-round interactive proof systems for all languages in coNP [C. Lund, L. Fortnow, H. Karloff, N. Nisan, Algebraic methods for interactive proof systems, Journal of the Association for Computing Machinery 39 (4) (1992) 859–868]. We consider the question of whether it is possible for coNP to have interactive proof systems with polylogarithmic-round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system, then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp–Lipton theorem and Yap’s theorem.