Polylogarithmic Round Arthur-Merlin Games and Random-Self-Reducibility

We consider Arthur-Merlin proof systems where (a) Arthur is a probabilistic quasi-polynomial time Turing machine, denoted AMqpoly, and (b) Arthur is a probabilistic exponential time Turing machine, denoted AMexp . We prove two new results related to these classes. We show that if co-NP is in AMqpoly then the exponential hierarchy collapses to AMexp. We show that if SAT is polylogarithmic round adaptive random-self-reducible, then SAT is in AMqpoly with a polynomial advice. The first result improves a recent result of Selman and Sengupta (2004) who showed that the hypothesis collapses the exponential hierarchy to S 2 .P; a complexity class which contains AMexp. The second result implies that if SAT is polylogarithmic round adaptive random-selfreducible, then the exponential hierarchy collapses. This partially answers a question posed by Feigenbaum and Fortnow (1993) who showed that if SAT is logarithmic round adaptive random-self-reducible then the polynomial hierarchy collapses.