  Let $m\geq 3$, we prove that $(\alpha n^\theta \mod 1)_{n>0}$ has Poissonian
$m$-point correlation for all $\alpha>0$, provided $\theta<\theta_m$, where
$\theta_m$ is an explicit bound which goes to $0$ as $m$ increases. This work
builds on the method developed in Lutsko-Sourmelidis-Technau (2021), and
introduces a new combinatorial argument for higher correlation levels, and new
Fourier analytic techniques. A key point is to introduce an `extra' frequency
variable to de-correlate the sequence variables and to eventually exploit a
repulsion principle for oscillatory integrals. Presently, this is the only
positive result showing that the $m$-point correlation is Poissonian for such
sequences.
