PokéBase - Pokémon Q&A
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Self-Explanatory.

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No.
Although not stated in Bulbapedia or this website, I tested this on Pokemon Showdown and after I Baton Passed my Protosynthesis to Mew (no protosynthesis), nothing happened. No boosts. And then, in the next game I Baton Pass to Flutter Mane (yes protosynthesis) but still no boosts.

Since Protosynthesis is the past paradox equivalent to Quark Drive, we can safely assume that this same logic applies to Quark Drive too.

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Thanks for that!
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No. Baton Pass only passes stat stages; the stat boosts from Protosynthesis and Quark Drive are passive effects of those abilities.

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Baton pass also passes other status effects, like substitute and leech seed. How do you know that protosynthesis and quark drive aren't 2 of those other effects?
@Vileplume Do you want this question back on the unanswered list?
Pretty safe to say that this guy didn't test anything.