Can time be measured without using entropy? All types of clocks I know of, are based on some entropy system (oscillating quartz crystal, spring , pendulum etc.). Is there any other way of measuring time then this type of systems?
 A: Any measurement of time involves measuring change of some type (level of water in jar, number of oscillations of pendulum, number of vibrations of quartz crystal etc.). The change has to be non-reversible otherwise the clock could just as easily run backwards as forwards - which is part of the reason why pendulum clocks have an escapement. The second law of thermodynamics tells us that a non-reversible change involves an increase in entropy.
A: Without entropy increasing there can still be time, just without a single arrow of time or only the second law. The example Julian Barbour often brings up is to imagine a 3 particle Newtonian system, 2 of which are a Kepler pair traveling along an axis in one direction and the remaining traveling toward the COM of the Kepler pair along the same axis.
The result after meeting is the single particle replaces one of the pair particles, and the new Kepler pair now moves off along a perpendicular axis, and the newly single particle along the same perpendicular axis in the opposite direction.
Now imagine a whole system of many of these 3 particle systems colliding, exchanging, traveling, then colliding, and so on.
In this idealized universe, there is no second law, but there is still time. The pairs are orbiting, exchanging momentum is happening, velocities are changing, and every snapshot is different from the next - hence time.
If you focus on one of the 3 particle systems, a pair and a single coming together then swapping, then going in their new directions, you will see reversing the system is a completely valid solution. No arrow is favored. If you look at the "shape" the 3 particles make - the Kepler pair and an incoming single - at first it is very thin isosceles triangle, nearly a line, with each Kepler pair a vertex of the base and the single as the other distant vertex. As they come together this shape changes toward an equilateral triangle, then the 3 interact and the pair and the new single go off in opposite directions along a perpendicular axis to the original. The triangle is now approaching a line again as they get further apart.
Julian calls the equilateral triangle portion a janus point. From the janus point the equilateral triangle has two arrows of time in opposite directions. Each going from equilateral toward isosceles toward a line, then back toward a new isosceles then equilateral as they encounter new pairs and singles.
This is time without the second law. It is an alternative to the "past hypothesis", that the entire universe was in an extremely low entropy state. We assume the past hypothesis to construct the second law. It is a necessary condition for having only a single arrow of time, namely the second law. Julian is obviously not in favor of that view an presents this janus point/shape dynamics alternative.
Note even with the second law, reversing every trajectory is a valid solution to our laws of physics. We just don't see that on large scales because of the past hypothesis.
I would recommend these videos of Julian's to clear up any lacking in my description.
https://youtu.be/K5eR8dHVyHk?t=994
https://youtu.be/f29y1IzWwwI?t=1860
