  We investigate the level spacing distribution for the quantum spectrum of the
square billiard. Extending work of Connors--Keating, and Smilansky, we
formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for
sums of two squares, and show that it implies that the spectral gaps, after
removing degeneracies and rescaling, are Poisson distributed. Consequently, by
work of Rudnick and Uebersch\"ar, the level spacings of arithmetic toral point
scatterers, in the weak coupling limit, are also Poisson distributed. We also
give numerical evidence for the conjecture and its implications.
