  General relativity predicts the existence of closed timelike curves (CTCs),
along which an object could travel to its own past. A consequence of CTCs is
the failure of determinism, even for classical systems: one initial condition
can result in multiple evolutions. Here we introduce a new quantum formulation
of a classic example, where a billiard ball can travel along two possible
trajectories: one unperturbed and one, along a CTC, where it collides with its
past self. Our model includes a vacuum state, allowing the ball to be present
or absent on each trajectory, and a clock, which provides an operational way to
distinguish the trajectories. We apply the two foremost quantum theories of
CTCs to our model: Deutsch's model (D-CTCs) and postselected teleportation
(P-CTCs). We find that D-CTCs reproduce the classical solution multiplicity in
the form of a mixed state, while P-CTCs predict an equal superposition of the
two trajectories, supporting a conjecture by Friedman et al. [Phys. Rev. D 42,
1915 (1990)].
