How to check satisfiability of a large number of “lorenzian” quadratic inequalities

Given a list of $$m$$ vectors $$x^i=(x^i_t,\textbf{x}^i)\in\mathbb{R}^{n+1}$$, $$i\in \mathbb{Z}_m$$ and two disjoint sets of vector pairs $$A,B\subset \mathbb{Z}_m\times\mathbb{Z}_m$$ as well as a set $$C\subset \mathbb{Z}_m\times\mathbb{Z}_m$$ is there an easy way (computational or analytical) to check if the following set of inequalities is satisfiable?

$$\forall(i,j)\in A\quad (x_t^i-x_t^j)^2-(\textbf{x}^i-\textbf{x}^j)^2>0$$

$$\forall(i,j)\in B\quad (x_t^i-x_t^j)^2-(\textbf{x}^i-\textbf{x}^j)^2<0$$

$$\forall (i,j)\in C\quad x_t^i-x_t^j>0$$