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Denote by $\mathcal{R}_p$ the set of all quadratic residues in $\mathbf{F}_p$for each prime $p$. A conjecture of A. S\'ark\"ozy asserts, for allsufficiently large $p$, that no subsets$\mathcal{A},\mathcal{B}\subseteq\mathbf{F}_p$ with$|\mathcal{A}|,|\mathcal{B}|\geqslant2$ satisfy$\mathcal{A}+\mathcal{B}=\mathcal{R}_p$. In this paper, we show that if suchsubsets $\mathcal{A},\mathcal{B}$ do exist, then there are at least $(\log2)^{-1}\sqrt p-1.6$ elements in $\mathcal{A}+\mathcal{B}$ that have uniquerepresentations and one should have \begin{align*} \frac{1}{4}\sqrt{p}<|\mathcal{A}|,|\mathcal{B}|< 2\sqrt{p}-1. \end{align*} This refines previousbounds obtained by I.E. Shparlinski, I.D. Shkredov, and Y.-G. Chen and X.-H.Yan. Moreover, we also establish bounds for $|\mathcal{A}|,|\mathcal{B}|$ andthe additive energy $E(\mathcal{A},\mathcal{B})$ if few elements in$\mathcal{A}+\mathcal{B}$ have unique representations.

A conjecture of S\'ark\"ozy on quadratic residues, II