How can I activate my iPhone5S after Restore?
It does not accept my current Apple ID and asks for the one that was used to set up my iCloud account, which I do not have
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If you mean "activation lock", neither we (just users like you) nor Apple can help you, sorry, unless you are the original owner and you still have the proof of purchase (receipt/invoice).
Shouldn’t be this case, activation lock can be removed only by the previous owner, with the credentials that were used to activate that iPhone. Without those credentials, that device is useless to you.
Regards
Giulio