Chowla's Construction of prime having least quadratic non-residue $\gg \log p$ This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the
 first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues".
I recently found a proof of this here.
The proof constructs such primes using this idea:


*

*Consider numbers which satisfies these condition : $n=1\bmod 8$ and $n=1\bmod r_i$ where $r_i$ are first consecutive odd primes less than $y$.

*Let $R:= 8\prod r_i $ , clearly $n=1\bmod R$ by Chinese remainder theorem.

*By Linniks there exist a prime $p$ in this arithmetic progression $\{ 1+rk_j\}$ such that $p= \mathcal{O}(R^6)$.

*Note that for this prime all numbers less than $y$ are quadratic residue, since p $\equiv  1 \bmod 8 \implies \big(\frac{a}{p} \big) = \big( \frac{p}{a} \big) $.

*By prime number theorem it is easy to see that y = $O(\log p)$  


But this analysis clearly doest rule out the possibility of having least quadratic non-residue $O(\log p)$ . Infact if we write all the constants and work it out than it doesn't even rule out the possibility of least quadratic non residue less than $10\log p$.
I want to know is this just what Chowla proved or there is a better argument for this. Any refrence for the same would be of great help.  

Since I am writing as an answer, I will put more information than the comment. 
Graham and Ringrose proved that:

There are infinitely many primes $p$ such that 
  $$
n_p\gg \log p \log\log\log p.$$

Montgomery proved that: 

If the Generalized Riemann Hypothesis(GRH) is true, then there are infinitely many primes $p$ such that 
  $$n_p\gg \log p \log\log p.$$

