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.