We focus on two particular models of closed timelike curves.
The first is Deutsch's 1991 mixed-state-fixed-point model which abolishes the grandfather paradox. The second is due to Bennett and Schumacher, in which the grandfather paradox on certain initial states is exploited as a putative physical mechanism for post-selection.
We discuss possible consequences of the existence of such closed timelike curves on quantum information processing.
Joint work with Charles Bennett, Graeme Smith, and John Smolin from IBM TJ Watson Research Center