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By the time we invent any sort of lightspeed travel, we’ll have long conquered quantum entanglement. If you have a signal transferred over a properly quantum entangled technology, the signal would transfer instantaneously.
Another option would be tiny temporary Einstein Rosen bridges. Sure the energy requirements would be hideous, but if we’ve figured out how to exceed C, I don’t think we really care about energy costs anymore.
You cannot transmit information through entangled particles, so probably not.