First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP^cc, and a version with unrestricted bias called UPP^cc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PP^cc \varsubsetneq UPP^cc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials.

Second, we study how small the bias of minimal-degree polynomials that sign-represent Boolean functions needs to be. We show that the worst-case bias is at worst double-exponentially small in the sign-degree (which was very recently shown to be optimal by Podolski), while the averagecase bias can be made single-exponentially small in the sign-degree (which we show to be close to optimal).