Privacy and mutual authentication under corruption with temporary state disclosure are two significant requirements for real-life applications of RFID schemes. No RFID scheme is known so far to meet these two requirements. In this paper we propose two practical RFID schemes that fill this gap. The first one achieves destructive privacy, while the second one narrow destructive privacy, in Vaudenay's model with temporary state disclosure. Both of them provide mutual (reader-first) authentication. In order to achieve these privacy levels we use Physically Unclonable Functions (PUFs) to assure that the internal secret of the tag remains hidden against an adversary with invasive capabilities. Our first RFID scheme cannot be desynchronized for more than one step, while the second one avoids the use of random generators on tags. Detailed security and privacy proofs are provided.