Variance of square-full integers in short intervals and arithmetic progressions

Define a natural number $n$ as a $\mathit{\text{square-full}}$ integer if for every prime $p$ such that $p\mathrm{\mid }n$, we have $p^2\mathrm{\mid }n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected order, under the assumption of a certain quasi-Riemann hypothesis. We also prove an asymptotic formula for the variance in arithmetic progressions, averaging over a quadratic residue and a nonresidue by a half, which is of smaller order of magnitude than the aforementioned bound for all primes $q\gg x^{51\mathrm{/}114+\epsilon }$.