Abstract: We formulate a conjecture, analogous to the Hardy--Littlewood prime tuples conjecture, for tuples of sums of two squares. Assuming this conjecture holds, we show that sums of two squares are distributed among the natural numbers as if according to a Poisson process. We discuss numerical evidence, and unconditional results, in support of our conjecture. This is joint work with Pär Kurlberg and Lior Rosenzweig.