Can entangled particles stay in sync when one of them is travelling near light speed? Can entangled particles stay in sync when one of them is travelling near light speed? I ask this because it means that time would be experienced at different speeds for both particles and could allow information about the past to stay in sync with information about the present if so
 A: Entanglement means that the particles involved are in a quantum mechanical state where all the phases and values are known.
Take a specific decay of one particle into two. In the case of the pi_0 two photons come from the decay and their spins are "entangled" because pi_0 has zero spin and angular momentum has to be conserved. Both photons go at the velocity of light.
The decay of a specific lamda0 particle into a pi- and a proton will have the particles entangled also. If the lamda has a high energy one of the products could be fairly close to the velocity of light. In the center of mass system of the lamda the momenta are always limited from conservation of energy and momentum.

time would be experienced at different speeds for both particles and could allow information about the past to stay in sync with information about the present if so

I do not understand this statement.
A: Entangled particles simply do not have the information you think they have.
Consider the simplest case, a particle that could spin up, down, or be in a superposition of both (this truly is as simple as it gets).
If you have two particles, one possibility is that they each have those properties.  For instance the first one could be up, the second could be up.  Another possibility is that both are down.  In both of those cases they are 100% correlated, but they are also 100% not entangled.  They are not entangled because the correlation is accidental, you could put one particle into a machine that turns particles upside down, and after it comes out of the machine, the two particles are now 100% anticorrelated.
When you entangle particles, they give up having their own individual properties now in favor of having their properties later be correlated.  So for instance you could have a super position of up-up and down-down.  This combination is an entanglement since we don't know if the particle will be up or down if we measure them, but we know that if we measure both they will be the same.  So now if we stick just one particle into a machine that flips it upside down, we don't change the correlation between the two since the one particle by itself was up just as much as it was down so it becomes down just as much as it was up.  Effectively its like the spin of both are going into the machine whenever either goes into the machine. And it's not just that machine, everything sees the pair of correlated spins, and has to act on the pair in a way that preserves the correlation.  But it is a correlation of things that later will be individually random.
Entangled particles become entangled by giving up their ability to have an isolated well defined property in favor of having properties that will be correlated under joint measurements.
So they don't have individual memories of what's happened to them.  They have indeterminate states that are setup to become correlated when measured.  After they are measured we learn about them as individuals, but since they didn't have individual properties in the past, we can't use the individual property now to learn about what happened in the past.
If it was just up and down, and the process of flipping upside down, it might seem like we can find out what it was in the past, but in reality there are many different experiments you can due (and that is why we have to have superpositions of up and down to describe all the ways it could act) and to describe what happens in all the possible experiments will require more information than just "it was up, or "it was down".
