ps={(a^2),a (1-a),(((1-a))^2),a(1-a),a(1-a),(((1-a))^2),a(1-a),(a^2)}; ps = ps/2;
vtp1={1-p,1-p,-1-p,-1-p,1+p,1+p,-1+p,-1+p};
vtp2={1-p,-1-p,1+p,-1+p,1-p,-1-p,1+p,-1+p};

gamma1=Apply[Plus,ps vtp1 vtp2]//Simplify

Solve[gamma1==0,p]

Apply[Plus,ps vtp1 vtp1]//Simplify
Apply[Plus,ps vtp2 vtp2]//Simplify

Apply[Plus,ps vtp1 (vtp2^3)/.{p->-1+2 a}]//Simplify