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We consider a generalization of the standard oracle model in which the oracleacts on the target with a permutation selected according to internal randomcoins. We describe several problems that are impossible to solve classicallybut can be solved by a quantum algorithm using a single query; we show thatsuch infinity-vs-one separations between classical and quantum querycomplexities can be constructed from much weaker separations.  We also give conditions to determine when oracle problems---either in thestandard model, or in any of the generalizations we consider---cannot be solvedwith success probability better than random guessing would achieve. In theoracle model with internal randomness where the goal is to gain any nonzeroadvantage over guessing, we prove (roughly speaking) that $k$ quantum queriesare equivalent in power to $2k$ classical queries, thus extending results ofMeyer and Pommersheim.

Uselessness for an Oracle Model with Internal Randomness