  The theory of relativity associates a proper time with each moving object via
its world line. In quantum theory however, such well-defined trajectories are
forbidden. After introducing a general characterisation of quantum clocks, we
demonstrate that, in the weak-field, low-velocity limit, all "good" quantum
clocks experience time dilation as dictated by general relativity when their
state of motion is classical (i.e. Gaussian). For nonclassical states of
motion, on the other hand, we find that quantum interference effects may give
rise to a significant discrepancy between the proper time and the time measured
by the clock. The universality of this discrepancy implies that it is not
simply a systematic error, but rather a quantum modification to the proper time
itself. We also show how the clock's delocalisation leads to a larger
uncertainty in the time it measures -- a consequence of the unavoidable
entanglement between the clock time and its center-of-mass degrees of freedom.
We demonstrate how this lost precision can be recovered by performing a
measurement of the clock's state of motion alongside its time reading.
