  We derive the predicted time dilation of delocalized atomic clocks in an
optical lattice setup in the presence of a gravitational field to leading order
in quantum relativistic corrections. We investigate exotic quantum states of
motion whose relativistic time dilation is outside of the realm of classical
general relativity, finding a regime where $^{24}\mathrm{Mg}$ optical lattice
clocks currently in development would comfortably be able to detect this
quantum effect (if the technical challenge of generating such states can be met
and the expected accuracy of such clocks can be attained). We provide a
detailed experimental protocol and analyse the effects of noise on our
predictions. We also show that the magnitude of our predicted quantum
relativistic time dilation effect remains just out of detectable reach for the
current generation of $^{87}\mathrm{Sr}$ optical lattice clocks. Our
calculations agree with the predicted time dilation of classical general
relativity when restricting to Gaussian states.
