Hard properties with (very) short Pcpps and their applications

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed `, we construct a property P(`) ⊆ {0, 1}ⁿ satisfying the following: Any testing algorithm for P(`) requires Ω(n) many queries, and yet P(`) has a constant query PCPP whose proof size is O(n · log(`) n), where log(`) denotes the ` times iterated log function (e.g., log<sup>(2)</sup> n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n · polylog n). As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed `, we construct a property that has a constant-query tester, but requires Ω(n/log(`)(n)) queries for every tolerant or erasure-resilient tester.