The circle shown has center $O$, circumference $120\pi$, and diameters $\overline{PR}$ and $\overline{QS}$. The length of arc $\overline{PQ}$ is $20\pi$. The length of arc $\overline{PS}$ is twice the length of arc $\overline{PQ}$. What is the length of arc $\overline{QR}$?