For a prime $p$ denote by $r(p)$ (resp. $n(p)$) the smallest prime $q$ which is a quadratic residue (resp. nonresidue) modulo $p$. It was shown by Linnik that for any fixed $\epsilon>0$ the number of $p<x$ s.t. $n(p)>p^\epsilon$ is bounded by $c(\epsilon)\log\log x$.
My question is what is the best known bound today and what is the best known corresponding bound for $r(p)$.
Edit: I mean a bound on the number of exceptions to Vinogradov's conjectures, i.e. $n(p),r(p)>p^\epsilon$.
There is the following result of Wolke from $1967$ (which is perhaps not the best, but quite good). Theorem: Let $p$ be an odd prime, and $L(s,\chi)$ the $L$-series for the Dirichlet character $(n/p)$. If $t(p)$ is a positive function with $L(1,\chi)>t(p)/\log(p)$, then there are absolute constants $c_1,c_2>0$ with $$ r(p)\le c_1 p^{c_2/\sqrt{t(p)}}. $$ Here one should mention the result of Elliott: if we have for an integer $k\ge 0$ and real $c>0$ $$ L(1,\chi)\ge \frac{c(\log \log p)^k}{\log p}, $$ then for every $\epsilon >0$ we have $r(p)\le c(\epsilon)p^{1/4(1+\epsilon)(k+2)^{-1}}$.
For $n(p)$ see the the report of Terence Tao, and http://www.math.ubc.ca/~gerg/teaching/613-Winter2011/LeastQuadraticNonResidue.pdf.
