I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have \begin{align*} \mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}x_ke^{2\pi i u k^2}\Big|\ge (\log n)^\alpha\Big\}\le \frac{1}{n^\beta} \end{align*} for some $\alpha>0$ and $\beta>0$. Note that $x$ is a fixed vector. Larger $\beta$ and smaller $\alpha$ is of course better.

Please note that I do not want this result for all $x$. Just for a fixed x. That is we have a fixed x and then draw the random variable $u$.