Let $n$ be a positive integer. How large must be a set $A\subset F_2^n$ to ensure that if $P$ is a quadratic polynomial in $n$ variables, vanishing at all non-zero points of the sumset $2A:=\{a_1+a_2\colon a_1,a_2\in A\}$, then also $P(0)=0$?

Considering the situation where $P(x_1,\ldots,x_n)=\sum x_ix_j+\sum x_i+1$, and $A$ consists of $0$ and the elements of the standard basis, we see that having $|A|>n$ is not enough. On the other hand, if $|A|>C2^{3n/4}$ with an appropriate absolute constant $C$, then $A$ contains a $3$-dimensional affine subspace, whence $2A$ contains a $3$-dimensional linear subspace, and the assertion is easy to deduce. Where exactly between $n$ and $2^{3n/4}$ lies the truth?