I will explain a construction of the motivic filtration on the topological cyclic homology of ring spectra, generalizing work of Bhatt–Morrow–Scholze and Bhatt–Lurie on the topological cyclic homology of discrete rings. This is joint with Arpon Raksit and Dylan Wilson. As time permits, I will discuss works in progress using the motivic spectral sequence to obtain new calculations in algebraic $K$-theory and prove higher chromatic variants of local Tate duality.[-]