  We study hypothesis testing under communication constraints, where each
sample is quantized before being revealed to a statistician. Without
communication constraints, it is well known that the sample complexity of
simple binary hypothesis testing is characterized by the Hellinger distance
between the distributions. We show that the sample complexity of simple binary
hypothesis testing under communication constraints is at most a logarithmic
factor larger than in the unconstrained setting and this bound is tight. We
develop a polynomial-time algorithm that achieves the aforementioned sample
complexity. Our framework extends to robust hypothesis testing, where the
distributions are corrupted in the total variation distance. Our proofs rely on
a new reverse data processing inequality and a reverse Markov inequality, which
may be of independent interest. For simple $M$-ary hypothesis testing, the
sample complexity in the absence of communication constraints has a logarithmic
dependence on $M$. We show that communication constraints can cause an
exponential blow-up leading to $\Omega(M)$ sample complexity even for adaptive
algorithms.
