Large values of short character sums

It is well-known that sums of values of non-principal Dirichlet characters over large enough intervals can be estimated non-trivially. However, it turns out that for any $A>0$ there exist infinitely many primes $p$ for which there is no non-trivial bound for the sum of values of quadratic character modulo $p$ over the interval of length $(\log p)^A$. We will discuss the proof of this result and its possible generalizations.