  We describe all sets $A \subseteq \F_p$ which represent the quadratic
residues $R \subseteq \F_p$ as $R=A+A$ and $R=A\hat{+} A$. Also, we consider
the case of an approximate equality $R \approx A+A$ and $R \approx A\hat{+} A$
and prove that $A$ has a structure in the situation.
