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What would be the most correct way to write this sentence?
"Everyone must remain quite in the hallways at all times.
Or
"Everyone should refrain from talking in the hallways at all times."
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The first is better, simpler, and more direct.

The only thing wrong is a typo: "quiet" is misspelled "quite".

Cheers

John.
Comments  
And: "everyone should refrain from talking in the hallways at all times" could be construed to mean that one may talk in the hallway some times, just not "all times".