What is the latest science on closed timelike curves? In Scientific American (Sept 2014), Lee Billings writes:

Lloyd, though, readily admits the speculative nature of CTCs. “I have no idea which model is really right. Probably both of them are wrong,” he says. Of course, he adds, the other possibility is that Hawking is correct, “that CTCs simply don't and cannot exist." Time-travel party planners should save the champagne for themselves—their hoped-for future guests seem unlikely to arrive.

What authoritative physicists say closed timelike curves (CTCs) exist?
Further to a comment, here is an article about an experiment apparently using post-selection closed timelike curves, (P-CTCs)

Because P-CTCs are based on post-selected teleportation, their predictions can be experimentally demonstrated. To experimentally demonstrate the grandfather paradox, we store two qubits in a single photon: one in the polarization degree of freedom, which represents the forward-travelling qubit, and one in a path degree of freedom representing the backward travelling qubit as shown in Fig 3.
... probe qubits measure the state of the polarization qubit before and after the quantum gun is “fired”. When the post-selection succeeds (i.e. the time travel occurs), the state of the probe qubits is measured.

 A: The Thorne time machine, a wormhole with one opening accelerated or Lorentz boosted outwards and then conversely brought back, does not permit time travel prior to the Cauchy horizon. This is the point where the time machine is "turned on." This Cauchy horizon has in regions of spacetime prior to its formation a set of curves winding through the wormhole that pile up towards it. In the future of its formation there are negative time directed curves that wind around the wormhole and pile up in the opposite time direction. This horizon is then a type of singularity, called a Cauchy horizon because it is limit as a Cauchy sequence of curves. 
Does this exist? It might in a quantum mechanical sense exist. This can only occur if there is a violation of the Hawking-Penrose energy conditions. A quantum vacuum can be squeezed so the uncertainty in conjugate variables is off quadrature. This would the case if $\Delta x~\rightarrow~0$ and $\Delta p~\rightarrow~\infty$. So this physics might play a role in quantum gravity. we might ponder whether the inner horizon of a Kerr-Newman black hole, that has Cauchy sequence properties, for a quantum black hole might squeeze the quantum gravity vacuum. It could be that quantum gravity permits a black hole, that is a nontraversable can quantum fluctuate into a traversable wormhole. This means there is a potential for CTCs in a "sum over histories" of a path integral.
Can this happen on a macroscopic scale? In other words, do wormholes have a classical correspondence? That Cauchy sequence piling of of curves would apply equally well to the vacuum, and it suggests at least that the time machine is unstable. There are reasons to think that a macroscopic wormhole is similarly unstable against any perturbation, which could include the vacuum. As yet we do not have a fundamental physics that presents the cosmic censorship and chronology protection hypotheses as derived or proven theorems. If these things might play a role in quantum gravity it is then at least interesting to ponder these questions.
Experimentally or from observation there is little evidence for large wormholes. the movie "Interstellar" had early on the observation of a wormhole, and the physics of its optical signature was in line with Thorne's papers on the subject. This of course requires a pretty close up observation, and we have as yet to image the region close to a black hole horizon. So these things are not ruled out, but they seem unlikely.
