Let $P$ denote the set of positive primes and let $p$ be a fixed prime. Then define $q\_{p}:=\min{q\in P:p<q,(q/p)=1}$ where $(⋅/p)$ is the Legendre symbol. So for instance $q\_{3}=7$, $q\_{3}=7$, and $q\_{5} = 11$. Is there anything known about $q\_{p}$ and specifically are there known bounds on $q\_{p}$ as a function of $p$? I am ultimately interested in investigating $\inf\{q\_{p}/p:p\in P \}$. 


I understand there are some things known about the smallest prime that is a quadratic residue modulo a prime, i.e. [least quadratic residue and nonresidue](https://mathoverflow.net/questions/137714/least-quadratic-residue-and-nonresidue), but this seems to be a different, albeit related, question. 



Smallest prime that is a quadratic residue modulo a fixed prime
Elaborating on Gerhard Paseman's idea, under a generalized Elliott-Halberstam conjecture the [Polymath8b paper](https://arxiv.org/abs/1407.4897) shows that there are infinitely many $n$'s such that at least two elements of $\{n,n+36,n+100\}$ are primes. In other words, assuming this conjecture, there are infinitely many prime pairs $(p,q)$ such that $q-p\in\{36,64,100\}$ is a bounded square, and hence
$$ \inf\{q/p:\ \text{$q>p$ are primes such that $(q/p)=1$}\}=1.$$


