A basic question in additive number theory is to study the sumset A+B for suitably arbitrary sets A,B with some prescribed structures. A conjecture of András Sárközy asserts, for all sufficiently large primes p, that no sumset A+B with |A|,|B|⩾2 consists of all quadratic residues mod p exactly. Sárközy himself proved the ternary analogue of this conjecture, and the original one seems beyond the current techniques. In this talk, we discuss some tight bounds for the possible binary decompositions, which are based on Weil’s bound for complete character sums over finite fields, improving some previous works by I. E. Shparlinski, I. D. Shkredov, and Y.-G. Chen and X.-H. Yan.This is joint work with Yong-Gao Chen.