We study the power of interactivity in local differential privacy. First, we focus on the difference between

We resolve this question. First, we classify locally private protocols by their \emph{compositionality}, the multiplicative factor k >= 1 by which the sum of a protocol's single-round privacy parameters exceeds its overall privacy guarantee. We then show how to efficiently transform any fully interactive k-compositional protocol into an equivalent sequentially interactive protocol with an O(k) blowup in sample complexity. Next, we show that our reduction is tight by exhibiting a family of problems such that for any k, there is a fully interactive k-compositional protocol which solves the problem, while no sequentially interactive protocol can solve the problem without at least an Omega(k) factor more examples.

We then turn our attention to hypothesis testing problems. We show that for a large class of compound hypothesis testing problems --- which include all simple hypothesis testing problems as a special case --- a simple noninteractive test is optimal among the class of all (possibly fully interactive) tests.