  Two parties observe independent copies of a $d$-dimensional vector and a
scalar. They seek to test if their data is correlated or not, namely they seek
to test if the norm $\|\rho\|_2$ of the correlation vector $\rho$ between their
observations exceeds $\tau$ or is it $0$. To that end, they communicate
interactively and declare the output of the test. We show that roughly order
$d/\tau^2$ bits of communication are sufficient and necessary for resolving the
distributed correlation testing problem above. Furthermore, we establish a
lower bound of roughly $d^2/\tau^2$ bits for communication needed for
distributed correlation estimation, rendering the estimate-and-test approach
suboptimal in communication required for distributed correlation testing. For
the one-dimensional case with one-way communication, our bounds are tight even
in the constant and provide a precise dependence of communication complexity on
the probabilities of error of two types.
