I have seen games using both forms time's up and time up to say that the time is up. Which is correct?
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They both mean the time is up, that is, the time allotted for this activity has expired. Time's up is a contracted form, and time up is an ellipted form. Which you use depends on context.
I think "time's up" is correct. It's a contraction for "time is up" (obviously). "Time up" seems to have no verb. It's just a noun and a preposition. Perhaps games that use "time up" are made in non-English-speaking countries. Foreign games often have slight missteps in colloquialisms.