Work of Bhatt--Morrow--Scholze defines a motivic filtration on the $p$-completed topological cyclic homology of discrete, quasisyntomic rings. In his ICM address, Rognes emphasized that his computations with Ausoni suggest a similar filtration on the algebraic $K$-theory of higher chromatic ring spectra. In this talk, I will explain how joint work with Arpon Raksit and Dylan Wilson rigorously defines the motivic filtration on the $p$-completed topological cyclic homology of chromatically quasisyntomic ring spectra. I will then explain work, variously of myself, Raksit, Wilson and Lee, that uses this motivic filtration as a practical computational tool.