Under SR, for two events (an event is a point in 4D space, i.e. a space and a time) with spacelike separation (i.e. outside each other's light-cones), neither comes objectively before or after the other. Different observers travelling at different speeds have different "surfaces of simultaneity", which say which events happen simultaneously. It's a theorem (or intuitively obvious) that for any two events with spacelike separation, there is a frame (i.e. a velocity at which you could be travelling) for which those two events are simultaneous.
So consider the events A=point 1, time 0; B=point 2, time 0; C= point 1, time 1 hour. As long as points 1 and 2 are more than 1 light-hour apart, it is legitimate to say that A is simultaneous with B, or that C is simultaneous with B. And so if you have a device that lets you travel from point 2 to point 1 in less than an hour (i.e. faster than light), you leave at B and arrive after A but before C. Then from the perspective of some observers (and the point of relativity is that all observers are equally legitimate), you arrived before you left.
The collapse of quantum entanglement can be confusing, which is part of why I prefer to follow many-worlds (i.e. no collapse postulate). You can construct quantum wavefunctions such that they're consistent with SR; for two entangled but spatially separated particles, you simply have a wavefunction that's a superposition of two states. When you as an observer observe one of those particles, you entangle yourself and your own state becomes part of the superposition. Of course "from the inside" it feels like you measured the particle and got one result or the other, but the actual wavefunction is just an ordinary superposition. From this perspective it's obvious no data was transmitted - you just became entangled with this distant particle without communicating with it, which is kind of odd, but no danger of violating causality.
But is it really important that it looks like you arrived after you left? Can't we just know that the light from one thing took longer to reach us than the other so what we are seeing is just an artifact of FTL.
Can that prevent causality? Or does it prevent relativity?
All observers are equally valid under SR; it would be very arbitrary to say that some of them are seeing "artifacts". Physics is based on observation, and the natural way to interpret our observations is to say that what we saw really happened. (Obviously this isn't true if some observable effect is distorting our perception, but that's not what's happening here - the observer who sees the person arrive before they left is a perfectly normal observer using perfectly normal equipment). Now if you want to say physics works differently for observers in different frames, that's fine, but it goes against relativity and all known physics.
Now assuming we accept that FTL-traveller really did arrive before they left, if that's all they do it's a matter of language as to whether they have violated causality by so doing. But it would take some very perverse physics (again, basically discarding relativity) to say they couldn't simply jump back, again going backwards in time, and arrive back at their starting point before they left it, at which you definitely have causality violation.
So consider the events A=point 1, time 0; B=point 2, time 0; C= point 1, time 1 hour. As long as points 1 and 2 are more than 1 light-hour apart, it is legitimate to say that A is simultaneous with B, or that C is simultaneous with B. And so if you have a device that lets you travel from point 2 to point 1 in less than an hour (i.e. faster than light), you leave at B and arrive after A but before C. Then from the perspective of some observers (and the point of relativity is that all observers are equally legitimate), you arrived before you left.
The collapse of quantum entanglement can be confusing, which is part of why I prefer to follow many-worlds (i.e. no collapse postulate). You can construct quantum wavefunctions such that they're consistent with SR; for two entangled but spatially separated particles, you simply have a wavefunction that's a superposition of two states. When you as an observer observe one of those particles, you entangle yourself and your own state becomes part of the superposition. Of course "from the inside" it feels like you measured the particle and got one result or the other, but the actual wavefunction is just an ordinary superposition. From this perspective it's obvious no data was transmitted - you just became entangled with this distant particle without communicating with it, which is kind of odd, but no danger of violating causality.