How much time it usually takes to xanax (alprazolam) to cause 'permanent' cognitive deficits like memory deficits?

Not permanent : Effects of xanax (alprazolam) are not permanent, can be reversible once it's stopped.

Not permanent: Any benzodiazepine can induce tolerance with ~2-3 weeks of consistent use. One might also have some temporary memory dysfunction with the first dose -- inability to learn new material when using. When the dose wears off, this effect should also wear off. If on xanax (alprazolam) and having this problem, a very gradual taper should allow cognition to return to previous abilities. It's not permanent.