Starting from Talagrand's inequality for quantum semigroups we will indicated why the Connes embedding problem appears naturally in the theory of tensor norms, more precisely Pisier vector-valued norms, and in matrix approximation of dynamics. Motivated by the connection to quantum information we will also formulate problems for multiple tensor products on C*-algebras and how they generalize Kirchberg's formulation of the Connes' embedding problem. This talk definitely has more questions then answers. This results are based on joint work with Li Gao, Nick LaRacuente, and Eric Ricard, Dimitry Shlyakhtenko.
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