I will show that a quadratic polynomial exhibits "random" behavior on the circle in the Poisson scaling regime. Specifically I will prove that second moments are Poissonian and motivate why other moments could be Poissonian. An essential feature of the polynomial under consideration is that it has a sufficiently "random" constant term. In the corresponding exponential sum the main term will not come from "major arcs" but from other points. Finally, I will relate this problem to gaps in $n^{1/3}\mod 1$ and show that Poisson moments (only two moments known at this time) for a specific quadratic polynomial imply exponential distribution for gaps.