Let $G$ be a group and $V$ a multiplicative abelian group. Then the cocycle condition is given by $$\delta\alpha(g_1,...,g_{k+1})=\alpha(g_1,...,g_k)^{(-1)^{k+1}}\alpha(g_2,...,g_{k+1})\prod_{i=1}^k\alpha(g_1,...,g_ig_{i+1},...,g_{k+1})^{(-1)^{i}}=1.$$
Let $G=\mathbb{Z}/n\mathbb{Z}$ and $V=U(1)$.
What would the condition be for $k=3$?
I think it is $$\delta\alpha(a,b,c,d)=\alpha(a,b,c)\alpha(b,c,d)\alpha(a+b,c,d)^{-1}\alpha(a,b+c,d)\alpha(a,b,c+d)^{-1}=1$$ Is this correct?