$V$ is a vector space over $\mathbb F_2$ of dimension $2n$. $B : V\times V \to \mathbb F_2$ is a nondegenerate symplectic form.
I'm reading a paper that states that there are essentially two quadratic forms $Q : V\to \mathbb F_2$ which have $B$ as their associated symplectic form. These are called $Q^+$ with $2^{2n-1} + 2^{n-1}$ roots, and $Q^-$ with $2^{2n-1} - 2^{n-1}$ roots.
I am trying to see why this is the case by using the properties of quadratic forms, but it's not clear to me why this leads to more than one quadratic form.
As a hint: consider $n=1$. The forms $Q^+$ and $Q^-$ are represented by $xy$ and $x^2+xy+y^2$.
In general the buzzphrase is Arf invariant.