A Hierarchy Theorem for Interactive Proofs of Proximity.

The number of rounds, or round complexity, used in an interactiveprotocol is a fundamental resource. In this work we consider thesignificance of round complexity in the context of InteractiveProofs 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, showingthat the power of IPPs grows with the number of rounds. Morespecifically, we show that there exists a gap functiong(r) = Theta(r^2) such that for every constant r geq 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 tu0027=poly(t)).In fact, we prove a stronger result by exhibiting a single language L such that, for every constant r geq 1, there is anO(r^2)-round IPP for L with t=n^{O(1/r)} verification time, whereas the verifier in any r-round IPP for L must run in time at least t^{100}. Moreover, we show an IPP for L with a poly-logarithmic number of rounds and only poly-logarithmic erification 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 standardinteractive proofs (in which the verifier can run in polynomialtime). Specifically, we show that the round reduction technique ofBabai 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).