  Let $\Delta_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the
$n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \|
e^{it \Delta_{\mathbb S^n}}f \|_{L^4([0, 2\pi) \times \mathbb S^n)} \leq C \|
f\|_{W^{\alpha, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $\alpha>
{(n-2)/4}.$ The range of $\alpha$ is sharp up to the endpoint. As a
consequence, we obtain space-time estimates for the Schr\"odinger propagator
$e^{it \Delta_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We
also prove that for zonal functions on ${\mathbb S}^n$, the Schr\"odinger
maximal operator $\sup_{0\leq t<2\pi} |e^{it\Delta_{\mathbb S^n}} f|$ is
bounded from $W^{\alpha, 2}(\mathbb S^n) $ to $L^{\frac{6n}{3n-2}}(\mathbb
S^n)$ whenever $\alpha>{1/3}$.
