  There is now a significant body of results on quantum interactions with
closed timelike curves (CTCs) in the quantum information literature, for both
the Deutsch model of CTC interactions (D-CTCs) and the projective model
(P-CTCs). As a consequence, there is a prima facie argument exploiting
entanglement that CTC interactions would enable superluminal and, indeed,
effectively instantaneous signaling. In cases of spacelike separation between
the sender of a signal and the receiver, whether a receiver measures the local
part of an entangled state or a disentangled state to access the signal can
depend on the reference frame. We propose a consistency condition that gives
priority to either an entangled perspective or a disentangled perspective in
spacelike separated scenarios. For D-CTC interactions, the consistency
condition gives priority to frames of reference in which the state is
disentangled, while for P-CTC interactions the condition selects the entangled
state. Using the consistency condition, we show that there is a procedure that
allows Alice to signal to Bob in the past via relayed superluminal
communications between spacelike separated Alice and Clio, and spacelike
separated Clio and Bob. This opens the door to time travel paradoxes in the
classical domain. Ralph (arXiv:1107.4675) first pointed this out for P-CTCs,
but we show that Ralph's procedure for a 'radio to the past' is flawed. Since
both D-CTCs and P-CTCs allow classical information to be sent around a
spacetime loop, it follows from a result by Aaronson and Watrous
(Proc.Roy.Soc.A, 465:631-647 (2009)) for CTC-enhanced classical computation
that a quantum computer with access to P-CTCs would have the power of PSPACE,
equivalent to a D-CTC-enhanced quantum computer.
