$0/1$ programming multiple quadratic constraints If we have an $n$-variable rank $n$-linear system it is clear we can find whether there exists a $0/1$ solution in polynomial time.
If we have an $n$-variable degree $2$ system how many constraints do we need to ascertain whether there exists a $0/1$ solution in polynomial time? What is the condition analogous to rank? Are there any standard algorithms? What is best known complexity?
 A: EDIT Arbitrary large number of equations doesn't guarantee you
polynomial time neither for the linear, nor for the quadratic system.
Take hard to solve linear or quadratic system.
Add arbitrary integer linear combinations of the equations.
You can add unbounded number of equations, but this does give
you any new information and doesn't change the rank of the linear
system.

Partial answer. Since you essentially detect unique linear solutions
and it is NP-complete, likely you can't hope for better for quadratic.
This is modification for overdetermined systems over finite fields.
Over $0/1$ we have $x^2=x$, so replace all $x^2$ with $x$, maybe giving
some linear terms.
Try "monomial linearization". Replace each $x_i x_j$ by new variable
$y_{ij}$ having in mind $x_i x_j=x_j x_i$ and keep the linear monomials.
This is linear system in at most $A=n(n-1)/2+n$ variables.
So if you have $A$ or more equations, you can hope for unique solution.
If unique solution exists over integers, first fix the linear variables
(if any).
If a $0/1$ solution exists, the unique solution must be of the form
$a y_{ij}=0$ or $a y_{ij}=a$ and the latter cases fixes $x_i,x_j=1$,
while in the former all non-fixed to $1$ variables are zero.
Likely this will work for randomly generated quadratic instances,
possibly showing no solution.
