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. $n(p),r(p)>p^\epsilon$.