Whatever it is, I don't believe paradoxes are possible (other than language ones that basically just confuse any attempts to resolve a statement or set of statements to true or false without breaking any physical laws or causality).

That said, I don't think an unstable time loop would necessarily be impossible. Eg, you go back in time and kill your grandfather before your father is conceived, which results in you never existing in the timeline, which then means no one is there to go back in time and kill your grandfather, which means the loop disappears and the timeline snaps back to the version where you do go back, and it continually alternates from there.

Not sure if any future outside of the unstable loop would exist, I think that would depend on if there's a higher dimension of time that these loops could play out over.

Or, if everything experiences the same present at the same time, it's also possible that after the first loop, it wouldn't go back to resolve the whole "killer pops out of literally nowhere" because it was in the past and no time traveler is bringing the timeline back to there, so it's all in the past. Though I think in that case, you wouldn't disappear after killing your grandfather. You'd just be an enigma that would require going outside of time to understand the origin of.

Tbh though I'm 99% sure time travel just isn't possible (paradoxes or not), just a fun thing to think about. And no, I don't consider quantum effects being symmetrical in time to be time travel, they are just cases where you can reverse cause and effect and still have a valid cause and effect sequence.