If the attacker has Meltdown/Specter-style access to your stack, register spills (especially but exclusively if not overwritten when you're done) can compromise

If the attacker has Meltdown/Specter-style access to your stack, register spills (especially but exclusively if not overwritten when you're done) can compromise your keys, but you need to have a rather weird threat model to avoid having them just read the actual keys out whatever storage you got them from in the first place.