The author considers the Poisson semigroup \(R(t)=e^{-t\sqrt{-\Delta-(n-1)P}}\) on the \(n\)-sphere. Here, \(\Delta\) is the Laplace-Beltrami operator on the \(n\)-sphere \(S_n\) and \(P\) is the projection operator onto spherical harmonics of degree \(\geq 1\). It is proven that \[ \|R(t)f\|_{L_p(S_n)}\leq \|f\|_{L_q(S_n)} \text{ for all }f \ \Leftrightarrow \ \ e^{-t}< \sqrt{\frac{p-1}{q-1}}. \] The last property is called hypercontractivity. Here, \(1<p\leq q<\infty\) and \(n\geq 1\).