Over the last century, the Hodge and Tate conjectures have inspired much activity in algebraic and arithmetic geometry. These conjectures give predictions for when certain topological objects come from geometry. Simpson and Fontaine-Mazur introduced non-abelian analogs of these conjectures. We prove these analogs for low rank local systems on generic curves, resolving conjectures of Esnault-Kerz and Budur-Wang as well as answering questions of Kisin and Whang.