Recently we have proven a noncommutative version of the extended De Finetti theorem for infinite sequences of random variables. These sequences canonically lead to endomorphisms of von Neumann algebras. In contrast to the classical result, the distributional symmetries of exchangeability and contractibility (aka ”spreadability“) are shown to be no longer equivalent in quantum probability. In joint work with R. Gohm, we have now identified an interesting class of contractable noncommutative random sequences. My talk will report some of our new results. We will show that braid group representations induce contractable noncommutative random sequences. We prove that these random sequences lead to triangular towers of von Neumann algebras, such that all cells form commuting squares. Our approach is applicable for nonhyperfinite von Neumann algebras and includes all examples from the Jones fundamental construction. We will illustrate our results by examples coming from the left regular representation of the braid group and the free group.