Quadratic residues and the combinatorics of sign multiplication

If S is a nonempty finite set of positive integers, we find a criterion both necessary and sufficient for S to satisfy the following condition: if q is a fixed nonnegative integer, then there exists infinitely many primes p such that S contains exactly q quadratic residues of p. This result simultaneously generalizes two previous results of the author, and the criterion used is expressed by means of a purely combinatorial condition on the prime factors of the elements of S of odd multiplicity.