We study the correlations of the celebrated Sine$_β$ point process. This point process arises as the bulk scaling limit of $β$-ensembles and has a geometric description through the Brownian carousel, as shown by Valkó and Virág (2009). We establish that the averaged $k$-point truncated correlation functions decay polynomially in the limit of large separation. We show that the decay exponent is of order $1/β$ for large $β$. This is a step towards a conjecture by Forrester and Haldane regarding the exact asymptotics of the two-point correlation function, a problem recently addressed by Qu and Valkó (2025). Our proofs, which rely on a careful analysis of the coupling of diffusions associated with the Brownian carousel, hold for all $β>0$ and $k \geq 1$, significantly extending previous results limited to specific values of $β$ or $k$.