We say that a set S is additively decomposed into two sets A and B if S = {a + b : a ∈ A, b ∈ B}. A. Sárközy has recently conjectured that the set Q of quadratic residues modulo a prime p does not have nontrivial decompositions. Although various partial results towards this conjecture have been obtained, it is still open. Here we obtain a nontrivial upper bound on the number of such decompositions. 2010 Mathematics Subject Classification. 11B13, 11L40.