I have a quantified convex program of the form that I need to solve

$$\exists(x_{1,1},\dots,x_{1,n})\in\mathbb R^n\quad\forall(x_{2,1},\dots,x_{2,n})\in\mathbb R^n$$ $$\vdots$$ $$\exists(x_{2t-1,1},\dots,x_{2t-1,n})\in\mathbb R^n\quad\forall(x_{2t,1},\dots,x_{2t,n})\in\mathbb R^n$$ $$\phi_1(x_{1,1},\dots,x_{2t,n})\leq a_1\wedge\dots\wedge\phi_r(x_{1,1},\dots,x_{2t,n})\leq a_r$$ where $\phi_1,\dots,\phi_r$ are either linear degree $d=1$ or quadratic degree $d=2$ convex polynomials with $O(n)$ terms each with all polynomial coefficients and $a_1,\dots,a_r$ in $\mathbb Z$ and having at most $m$ bits each.

I feel quantifier elimination over reals will help here. However how to go about it is unclear.

Does quantifier elimination tell anything about the time complexity of the program and if so what is the complexity and is it possible to use quantifier elimination to reduce the number of quantifications and if so how does the parameters change?

Is it in $\Sigma_{k}$ in the polynomial hierarchy where $k=O(t)$ and independent of $m,n,r$?