The unspoken thing about the Prime Directive is that a Federation Captain's most solemn duty is deciding when to ignore it, and the same goes for the Temporal Prime Directive.
Sure, but what about random crewmen, like in my example? Are they expected to make such a decision?
They should take it to their captain
As far as I know, the directive mainly applies to officers who are sent back in time and/or given the opportunity to change established history. I don't think it would prevent someone from making an arrest in their "proper" time.
At most, it might limit their ability to interrogate the prisoner, if they can verify that the intruder is from the future and possesses knowledge that the contemporary officers can't have.
I guess I assumed a sort of corollary.
Starfleet personnel ends up back in time on a Starfleet vessel. We both serve the same organization. My duty is to protect the timeline I come from. Your duty seems, implicitly, to aid a fellow Starfleet officer in their mission (to protect the aforementioned timeline).
It seems like Starfleet should have a dedicated Temporal Security crew on every starship and starbase for such an occasion. You find a supposed time traveler, you immediately call this team. They sequester the intruder and go through a careful interview to verify their claim as cleanly as possible, then render what aid is needed to secure the timeline and get them home (or, barring that possibility, get them somewhere isolated where they can't contaminate the timeline). Then, maybe memory wipe the Temporal Security team (and possibly anyone else who interacted with the traveler). On the flipside, if you end up back in time, it's expected you should immediately attempt to contact the local Temporal Security crew.
Chakotay once used the TPD as an excuse to not answer a question from Janeway.
And she just accepted it.
Now you’ve paused, thinking any action could cause a temporal paradox, or damage to the future timeline.
You can also arrest them anyway. Logically, it would fit into the timeline, since if they're from the future, they should have accounted for those circusmtances, including their arrest in their future timeline. And a scan would show a fair few forms of time travel, which you can use to verify, or any connections that you might have with other temporal entities. Time travel is a bit convenient like that, until everything goes horribly wrong.
But ultimately, nothing actually prevents someone from just out and out lying. They could easily say "oh, sorry, can't say, Starfleet Intelligence business", or "I'm here on Captain/Engineering's orders", and you'd have that delay in either case to verify the story. The Temporal Prime Directive doesn't really factor into it.
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