  We will investigate quasi-randomness for profinite groups. We will obtain
bounds for the mininal degree of non-trivial representations of
$\mathrm{SL}_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and
$\mathrm{Sp}_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Our method also delivers a
lower bound for the minimal degree of a faithful representation for these
groups. Using the suitable machinery from functional analysis, we establish
exponential lower and upper bounds for the supremal measure of a product-free
measurable subset of the profinite groups $\mathrm{SL}_{k}({\mathbb{Z}_p})$ and
$\mathrm{Sp}_{2k}(\mathbb{Z}_p)$. We also obtain analogous bounds for a special
subgroup of the automorphism group of a regular tree.
