Log Fano cone singularities are generalizations of affine cones over Fano varieties. Motivated by the study of canonical metrics on Fano varieties, there is a local K-stability theory characterizing the existence of Ricci-flat K\"ahler cone metrics on log Fano cone singularities. In this talk, we aim to give a non-Archimedean characterization of local K-stability. As an application, we will see that to test local K-semistability, it suffices to test special test configurations.