Stronger separations for random-self-reducibility, rounds, and advice

A function f is self-reducible if it can be computed given an oracle for f. In a random-self-reduction the queries must be made in such a way that the distribution of the ith query is independent of the input that gave rise to it. Random-self-reductions have many applications, including countless cryptographic protocols, probabilistically checkable proofs, average-case complexity, and program checking. A simpler model of randomized self-reducibility is coherence, in which the only condition on the queries is that the input itself may not be among the queries. We show that there is a function which is random-self-reducible with 2 rounds of queries, but which is not even coherent, even if polynomial advice is allowed, when the queries must be made in a single round