No, we have not. We have observed that we can generate a pair of particles and separate them, and when we look at the close one, we now know that the far one h

No, we have not. We have observed that we can generate a pair of particles and separate them, and when we look at the close one, we now know that the far one has the complementary property. You can't use that to send information. You could use it as a shared secret, but you still had to move the particle out where your recipient is for them to use it. You can take a flashlight and shine it at the moon, and if you sweep the beam back and forth, you can make the notional front of illumination move faster than the speed of light -- but you can't modulate the signal faster than the propagation velocity c. Time travel into the future is easy. Time travel into the past doesn't work in this universe.