Sumsets in quadratic residues

We describe all sets $A\subseteq \mathbb{F}_p$ which represent the quadratic residues $R\subseteq \mathbb{F}_p$ in the sense that $R=A+A$ or $R=A\hat{+}A$. Also, we consider the case of an approximate equality $R\approx A+A$ and $R\approx A\hat{+}A$ and prove that $A$ is then close to a perfect difference set.