  We study the power of interactivity in local differential privacy. First, we
focus on the difference between fully interactive and sequentially interactive
protocols. Sequentially interactive protocols may query users adaptively in
sequence, but they cannot return to previously queried users. The vast majority
of existing lower bounds for local differential privacy apply only to
sequentially interactive protocols, and before this paper it was not known
whether fully interactive protocols were more powerful. We resolve this
question. First, we classify locally private protocols by their
compositionality, the multiplicative factor $k \geq 1$ 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
$\tilde \Omega(k)$ 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.
