We prove that subexponential-time quantum Turing machines are superior to their classical counterparts within common space bounds in $Ω(\log \log n)$. For that purpose, we define infinitely many sets of ``padded palindromes'' that are distinguished from each other by the precise relationships between the lengths of the palindromic prefixes and the paddings. We exhibit an infinite family $\mathcal{F}$ of functions in $(\log n)^{ω(1)}\cap n^{o(1)}$ such that for every $\tilde{f}_{i}\in\mathcal{F}$, there exists another function $\tilde{f}_{i+1}\in\mathcal{F}$ such that $\tilde{f}_{i+1}(n) \in o(\tilde{f}_{i}(n))$, and each such $\tilde{f}_{i}$ corresponds to a different quantum advantage statement, i.e. a proper inclusion of the form $\mathsf{BPTISP}(2^{O(\tilde{f}_{i}(n))},o(\log \tilde{f}_{i}(n)))\subsetneq \mathsf{BQTISP}(2^{O(\tilde{f}_{i}(n))},o(\log \tilde{f}_{i}(n)))$ for a different pair of subexponential time and sublogarithmic space bounds. One can also obtain quantum advantage statements where the common space bound is $Θ(\log \log n)$ and the time bound is ``almost'' quasi-polynomial, i.e., of the form $2^{(\log n)^{Θ(f(n))}}$, where $f(n)\in ω(1)$ is a function that can be selected to grow very slowly. Our results depend on a technique enabling polynomial-time quantum finite automata to control the amount of padding with very fine asymptotic granularity.

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