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Magic coins are useful for small-space quantum machines

Although polynomial-time probabilistic Turing machines can utilize uncomputable transition probabilities to recognize uncountably many languages with bounded error when allowed to use logarithmic space, it is known that such "magic coins" give no additional computational power to constant-space versions of those machines. We show that adding a few quantum bits to the model changes the picture dramatically. For every language $L$, there exists such a two-way quantum finite automaton that recognizes a language of the same Turing degree as $L$ with bounded error in polynomial time. When used as verifiers in public-coin interactive proof systems, such automata can verify membership in all languages with bounded error, outperforming their classical counterparts, which are known to fail for the palindromes language.