  In a surprising recent work, Lemke Oliver and Soundararajan noticed how
experimental data exhibits erratic distributions for consecutive pairs of
primes in arithmetic progressions, and proposed a heuristic model based on the
Hardy--Littlewood conjectures containing a large secondary term, which fits the
data very well. In this paper, we study consecutive pairs of sums of squares in
arithmetic progressions, and develop a similar heuristic model based on the
Hardy--Littlewood conjecture for sums of squares, which also explain the biases
in the experimental data. In the process, we prove several results related to
averages of the Hardy--Littlewood constant in the context of sums of two
squares.
