"You have not performed any activity"
"You have not performed any action"
"There is no item in your activity tab"
Any other suggestions are also welcome.
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There is no activity to display
This works fine unless you want to underscore that it was the user who didn't perform any activity, as opposed to the system, for example.
From the three options you presented, "You have not performed any activity" is correct, too.
"You have not performed any action" should be "You have not performed any actions" (plural.)
"There is no item in your activity tab" should be "There are no items in your activity tab" (again, plural.)
But the last two are a bit inelegant, even after the grammar has been corrected.