I encountered the following value in my research:

Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$. Denote $$ L = \mathop{\mathrm{E}}_x[ \prod_{1\leq j\leq m} \langle \alpha_j,x \rangle^2], $$ where $x \in \mathbb{R}^n$ is a random vector whose $\ell$th component is i.i.d. uniformly over $\{-1,1\}$.

My observation is that if $m=n$, $L$ may be $0$. For instance, let $m=n=2$, $\alpha_1=\frac{1}{\sqrt{2}}(1\ 1)$ and $\alpha_2=\frac{1}{\sqrt{2}}(1\ {-1})$.
However, if, say, $m=o(n)$ or even $n/2$, then $L$ is *seemingly* lower bounded by $\Omega(1)$. How can I lower bound $L$ for those small $m$ (relative to $n$)?

For small and concrete $m$, I can manually lower bound $L$ by first expand the RHS and then apply the *random subsum principle* (namely, $\mathrm{E}_x[x_{\ell_1}x_{\ell_2} \dots x_{\ell_c}]=0$, where $x_{\ell_j}$ denotes the $\ell$th component of $x$). But I am not sure how to generalize this approach to arbitrary $m$.

Any hint or reference will be greatly appreciated (let alone an answer).

For those who are curious about the background:
I am working on *quantum query complexity*, and I am trying to use the *polynomial method* to solve some certain problems.
If you are interested but not familiar with these terms, refer to [BdW02].