Terry Tao gives this oblique definition of quasirandom group in his notes 3

$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at least $D$.

There is nothing random about this representation, and the size of these irreducible representations doesn't make it more random.

There is someting kind of mixing about it, I guess:

Let $G$ be a finite $D$ quasi-random group and $f,g \in \ell^2(G)$ such that $\mathbb{E}[f] = \mathbb{E}[g] = 0$ then $$ ||f \ast g||_{\ell^2(G)} \leq \frac{1}{\sqrt{D}} \; |G| \; \big|\big|f\big|\big|_{\ell^2(G)} \; \big|\big|g\big|\big|_{\ell^2(G)} $$

Not knowing any better, I'd say this looks somewhat like the Cauchy-Schwartz inequality and I'd expect to use it in the same way.

I don't know what it means to say a group $(G, \cdot)$ is quasi-random. Are there any combinatorial objects which are "quasi-random" associated to $G$ ?

In the case of $SL(2, \mathbb{F}_p)$ we find it is $\frac{p-1}{2}$-quasirandom. Can you randomly generate objects using the group $SL(2, \mathbb{F}_p$ ?