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In my impoverishing experience, getting a criminal record clearance letter from the FBI is much faster than the California DoJ but is more expensive (tested in Feb. 2021, using the "live scan fingerprinting" services in San Jose, California):
Also, beware that if some embassy is asking for a criminal record clearance letter from "a state", they might mean State Department (a.k.a. United States Department of State) and not your state DoJ. That's what happened in my case, and the embassy didn't bother to correct their website.