  We use the Burgess bound and combinatorial sieve to obtain an upper bound on
the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given
interval $[u+1,u+\psi(Q)]$ does not contain a quadratic non-residue modulo $p$.
The bound is nontrivial for any function $\psi(Q)\to\infty$ as $Q\to\infty$.
This is an analogue of the well known estimates on the smallest quadratic
non-residue modulo $p$ on average over primes $p$, which corresponds to the
choice $u=0$.
