Separations and equivalences between turnstile streaming and linear sketching

A longstanding observation, which was partially proven by Li, Nguyen, and Woodruff in 2014, and extended by Ai, Hu, Li, and Woodruff in 2016, is that any turnstile streaming algorithm can be implemented as a linear sketch (the reverse is trivially true). We study the relationship between turnstile streaming and linear sketching algorithms in more detail, giving both new separations and new equivalences between the two models.