  We provide a complete characterization of theories of tracial von Neumann
algebras that admit quantifier elimination. We also show that the theory of a
separable tracial von Neumann algebra $\mathcal{N}$ is never model complete if
its direct integral decomposition contains $\mathrm{II}_1$ factors
$\mathcal{M}$ such that $M_2(\mathcal{M})$ embeds into an ultrapower of
$\mathcal{M}$. The proof in the case of $\mathrm{II}_1$ factors uses an
explicit construction based on random matrices and quantum expanders.
