  Time has a fundamentally different character in quantum mechanics and in
general relativity. In quantum theory events unfold in a fixed time order while
in general relativity temporal order is influenced by the distribution of
matter. When the distribution of matter requires a quantum description,
temporal order is expected to become non-classical -- a scenario beyond the
scope of current theories. Here we provide a direct description of such a
scenario. We consider a massive body in a spatial superposition and show how it
leads to "entanglement" of temporal orders between time-like events in the
resulting space-time. This entanglement enables accomplishing a task, violation
of a Bell inequality, that is impossible under classical temporal order.
Violation of the inequality means that temporal order becomes non-classical --
it cannot be described by locally defined classical variables. Our approach
provides a quantitative method for investigating quantum aspects of space-time
and gravity.
