That still involves sending information faster than light, which still breaks relativity and can thus enable time travel and thus the possibility of temporal paradoxes. In real life we can't actually use quantum entanglement to transmit information FTL, it's just that any science fiction which
does use QE for FTL transmissions either handwaves away the causality breakage with more technobabble, or just flat-out ignores it.
Now of course, one is entirely free to suppose that something like the
chronology protection conjecture applies in a fictional universe, but there's no good reason why such a thing couldn't also apply to tachyons as well as quantum entanglement and warped spacetime. But it's just that; a conjecture. If FTL travel does turn out to be viable, then it may well be the case that causality is something that only applies below light speed. Which would have a number of implications that I am nowhere near well-read enough in theoretical physics to work out the consequences. But given how causality underpins pretty much all of known physics, it would be one of the greatest paradigm shifts in the entire history of science.
Alcubierre drives and wormholes don't strictly speaking enable FTL travel on a
local basis, but they most certainly do enable FTL travel on a
global basis. There's nothing theoretically stopping a ship with a warp drive from heading off to a distant star, and then heading back home to arrive before it left in the first place. Similarly, an arrangement of wormholes known as a
Roman ring can enable the FTL transmission of matter and information on a global basis.
Here's a great video by PBS Space Time that exposes the dirty great secret of FTL travel, namely that it allows time travel:
Source: https://www.youtube.com/watch?v=HUMGc8hEkpc