Periodicity in time is unusual to think about in physics, partly because you start to get wacky results. If you could establish a CTC in nature, it would allow for computers that can efficiently (i.e. in polynomial time) solve not just NP problems but actually even all of PSPACE [0]. You can interpret this in two ways. There’s the hopeful way: “we should spend a boatload of money trying to find CTCs in our universe since they’ll let us create super-ultra computers”. And then there’s the pessimistic way, that nature probably isn’t going to give us a free lunch like that. Sadly, I’m in the pessimist camp on this one!
[0] https://www.scottaaronson.com/papers/ctc.pdf