  For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo
$p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$.
This conjecture follows from the generalised Riemann hypothesis, and is known
to hold for almost all primes $p$ but remains open in general. In this paper we
show that Vinogradov's conjecture also follows from the Elliott-Halberstam
conjecture on the distribution of primes in arithmetic progressions, thus
providing a potential "non-multiplicative" route to the Vinogradov conjecture.
We also give a variant of this argument that obtains bounds on short centred
character sums from "Type II" estimates of the type introduced recently by
Zhang and improved upon by the Polymath project, or from bounds on the level of
distribution on variants of the higher order divisor function. In particular,
we can obtain an improvement over the Burgess bound would be obtained if one
had Type II estimates with level of distribution above $2/3$ (when the
conductor is not cube-free) or $3/4$ (if the conductor is cube-free); morally,
one would also obtain such a gain if one had distributional estimates on the
third or fourth divisor functions $\tau_3, \tau_4$ at level above $2/3$ or
$3/4$ respectively. Some applications to the least primitive root are also
given.
