Large scale work of many researchers worldwide has recently culminated in the classification of C*-algebras, which is now at the level of Connes' classification of injective factors. Nowadays, C*-algebras is at a similar state to that of von Neumann algebras after Jones introduced the index for subfactors in the early 80s. Thereafter, great interest has arisen in constructing and classifying UTC-actions on C*-algebras, aiming to understand their structure from the viewpoint of quantum symmetries.

We will see that every UTC $cC$ acts on some simple, unital separable and monotracial C*-algebra constructed only from $cC$ by adapting diagrammatic and free probabilistic techniques from GJS. Using a 'Hilbertification' technique, we recover the UTC-action constructed by Brothier-Harglass-Penneys of $cC$ on the free group factor $Lmathbb{F}_infty.$ This is joint work with Hartglass. Finally, we will review some recent developments and obstructions to the existence of UTC actions on (classifiable) C*-algebras.