Searches have returned absolutely no results on this question. With that in mind, I assume the answer is either painfully obvious ('of course quantum computers get no advantage when it comes to storing data') or the question has just not been studied. Intuitively it makes sense the quantum algorithms would gain no advantage related to storage, but memory hard functions seem to be a rather budding new technology and I wonder to what degree of certainty that intuition can be verified, if at all.
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