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.
See: NonDeterministic, NonDeterministicTuringMachine, QuantumComputersArentTuringEquivalent