We discuss various models of closed timelike curves (CTCs), and their
utility for increasing the distinguishability of nonorthogonal quantum states and
for speeding up hard computations. In the most widely used model,
Deutsch's 1991 mixed-state-fixed-point model, we show that CTCs are
of no help in quantum state discrimination and may not be of much help for
speeding up hard computations. (Joint work with Debbie Leung, Graeme Smith,
and John Smolin.) arXiv 0908.3023.