The possibility to non-deterministically explore an infinite number of possible branches (e.g. f(x) for all natural numbers x).

Such a device would allow algorithms or machines, that are beyond TuringEquivalent. They could e.g. solve the HaltingProblem and many other undecidable problems.
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See: NonDeterministic, NonDeterministicTuringMachine, QuantumComputersArentTuringEquivalent