Does this mean you could use a time crystal to efficiently measure time accurately, by counting the number of cycles? I know we do that with quartz oscillators but those need energy input to keep oscillating.
Good question. I was about to edit this into my comment, but now it works better as a reply.
The act of "reading" the configuration of a time crystal disturbs the time crystal. So you either a set of maximally-similar time crystals that you read at various times during a day, or you need to re-freeze your single disturbed time-crystal each time you read it.
There are ordinary crystals which literally melt out of their crystalline state when handled / measured-by-bright-light. The organized pattern is broken with the additional energy. Time crystals are patterened over time, and that pattern breaks when they are handled / measured-by-bright-light.
You could think of it as having to shine a flashlight (or laser) through the time crystal to figure out which way it twists the light at a given time t_x. If you know the temporally-periodic structure, you can predict the different twisting when you turn on the light at t_x versus t_x+1 or t_x-1. But lighting up the crystal breaks the lowest-energy condition of the time crystal -- it's melted by the light it twists -- so you have to re-freeze it back into its predictable periodic structure, otherwise you might get the same twisting (or none) at t_{measured}+1, t_{measured}+2, ..., t_{measured}+n.
(It is fairly literally re-freezing: you have to do laser cooling or the like. And it takes energy to run the cooler, which removes energy from the not-lowest-energy-state broken time crystal, so thermodynamics isn't violated.)
That's a very good question. I started but abandoned a fairly deep answer, mostly because this is an area far from my expertise and in which it is easy to be howlingly wrong. (To be fair to me, subject matter experts have been arguing about this in the literature for some twenty years.)
