A hierarchy theorem for interactive proofs of proximity

The number of rounds, or round complexity, used in an interactive protocol is a fundamental resource. In this work we consider the significance of round complexity in the context of Interactive Proofs of Proximity (IPPs). Roughly speaking, IPPs are interactive proofs in which the verifier runs in sublinear time and is only required to reject inputs that are far from the language. Our main result is a round hierarchy theorem for IPPs, showing that the power of IPPs grows with the number of rounds. More specifically, we show that there exists a gap function g(r) = θ(r²) such that for every constant r ≥ 1 there exists a language that (1) has a g(r)-round IPP with verification time t = t(n, r) but (2) does not have an r-round IPP with verification time t (or even verification time t0 = poly(t)). In fact, we prove a stronger result by exhibiting a single language L such that, for every constant r ≥ 1, there is an O(r2)-round IPP for L with t = nO(1/r) verification time, whereas the verifier in any r-round IPP for L must run in time at least t100. Moreover, we show an IPP for L with a poly-logarithmic number of rounds and only poly-logarithmic verification time, yielding a sub-exponential separation between the power of constant-round IPPs versus general (unbounded round) IPPs. From our hierarchy theorem we also derive implications to standard interactive proofs (in which the verifier can run in polynomial time). Specifically, we show that the round reduction technique of Babai and Moran (JCSS, 1988) is (almost) optimal among all blackbox transformations, and we show a connection to the algebrization framework of Aaronson and Wigderson (TOCT, 2009).