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I think: there is no difference .
sign-off sta must be post-layout sta. When layout is done , the spef parameter file is extracted and the sta is done to decide whether the layout is timing ok for tapeout.
what is the post-layout sta aim to do is also to determine whether the layout is timing ok for tapeout.
As advice from synopsys, sign-off STA should be four times STA in two corner, slow corner and fast corner.
You should run STA by primetime. Use bc-wc analysis mode.
Set slow library, check setup time and hold time.
Set fast library, check setup time and hold time.
If all above four conditions are timing closured. You can sign-off. That is recommend flow.
But commonly, you can read both slow library and fast library in and check setup time with slow and hold time with fast. That is fine as well
As for the SetUp and Hold checks, should I run the tool twice each time for either SetUp Or Hold check? If the tool is able to read slow and fast SDF together, why not running it only once for all the related checks?