A quantum permutation (or magic unitary) is given by a square matrix whose entries are self-adjoint projections acting on a common Hilbert space H with the property that the row and column sums each add up to the identity operator. Quantum permutations are operator-valued analogues of ordinary permutation matrices and they arise naturally in both quantum group theory and also in the study of quantum strategies for certain non-local games. From the perspective of non-local games, it is often of great importance to know whether or not a quantum permutation (possibly satisfying some additional algebraic relations among its entries) admits a matrix model. I.e., can it be realized via operators on a finite-dimensional Hilbert space H? In this talk, I will explain how in the case of ``generic'' quantum permutations, matrix models abound. More precisely, the universal unital ∗-algebra A(N) generated by the coefficients of an N×N quantum permutation is always residually finite dimensional (RFD). Our arguments are based on quantum group and subfactor techniques. As an application, we deduce that the II1-factors associated to quantum permutation groups satisfy the Connes Embedding Conjecture. This is joint work with Alex Chirvasitu and Amaury Freslon.