A quasi-Stone algebra is an algebra \((L,\vee,\wedge,',0,1)\) of type \((2,2,1,0,0)\) such that \((L,\vee,\wedge,0,1)\) is a bounded distributive lattice, \(0'=1\), \(1'=0\), \((x\vee y)'=x'\wedge y'\), \((x\wedge y')'=x'\vee y''\), \(x\leq x''\) and \(x'\vee x''=1\) for all \(x,y\in L\). A variety \(\mathcal V\) is called universal if every category of algebras of finite type is isomorphic to a full subcategory of \(\mathcal V\). If, in addition, there exists a functor \(\Phi\) from the category of simple graphs (with compatible mappings) to \(\mathcal V\) which establishes that \(\mathcal V\) is universal and if \(\Phi\) sends finite graphs to finite algebras then \(\mathcal V\) is called finite-to-finite universal. Certain varieties of quasi-Stone algebras are proved to be (not) finite-to-finite universal, not universal respectively (not) finite-to-finite relatively universal, a notion which is too complicated to be defined here.