The various electromagnetic versions of the Aharonv-Bohm effect can be described in a way that does not suggest any physical reality to the electromagnetic potentials and does not require any nonlocality, by treating the sources of the electromagnetic field and the electromagnetic field itself quantum mechanically, instead of treating the test charge (electron) quantum mechanically and the electromagnetic field and its sources classically. Without a theory of quantum gravity, we cannot appeal to a fully quantum treatment of the test mass (atom), the sources of the gravitational field, and the gravitational field itself in order to explain the gravitational Aharonov-Bohm effect in a fully local, gauge-independent manner. However, this effect can be explained in a local, gauge-independent manner by considering the gravitational time dilation experienced by the atom in the path with the nonzero potential, and taking into account that matter waves pick up a phase at the Compton frequency of the matter.