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The system is designed to allow your email provider or another Identity Provider that you trust, to store your password instead of Mozilla.

Your identity provider just has to implement the BrowserID protocol https://developer.mozilla.org/en-US/docs/Persona/Identity_Pr...

So what happens if my email account gets hacked? Won't this compromise all my accounts then?

This problem exists today. I can do "forgot my password" on many sites and owning your email account can change the passwords and log in to them.

Persona doesn't attempt to solve this existing problem.