Thurston conjectured that quasifuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a generalization of these conjecture to convex hulls of quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. (This is joint work in progress with Bonsante, Danciger and Schlenker.)