Simultaneously vanishing quadratic forms

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$:

$$c^* A_i c = 0?$$

Namely, when can the quadratic forms defined by the $\{A_i\}$ simultaneously vanish?

The system of equations you present is a quadratic system of polynomial equations for the components of $c$. As discussed here, you can compute the Gröbner basis for this system of equations, which will tell you if it is inconsistent or not. In order to exclude the zero vector as a solution you could add a normalization condition $c^* c = 1$. I think this should work.