  We analyze \emph{fractional Brownian motion} and \emph{scaled Brownian
motion} on the two-dimensional sphere $\mathbb{S}^{2}$. We find that the
intrinsic long time correlations that characterize fractional Brownian motion
collude with the specific dynamics (\emph{navigation strategies}) carried out
on the surface giving rise to rich transport properties. We focus our study on
two classes of navigation strategies: one induced by a specific set of
coordinates chosen for $\mathbb{S}^2$ (we have chosen the spherical ones in the
present analysis), for which we find that contrary to what occurs in the
absence of such long-time correlations, \emph{non-equilibrium stationary
distributions} are attained. These results resemble those reported in confined
flat spaces in one and two dimensions [Guggenberger {\it et al.} New J. Phys.
21 022002 (2019), Vojta {\it et al.} Phys. Rev. E 102, 032108 (2020)], however
in the case analyzed here, there are no boundaries that affects the motion on
the sphere. In contrast, when the navigation strategy chosen corresponds to a
frame of reference moving with the particle (a Frenet-Serret reference system),
then the \emph{equilibrium} \emph{distribution} on the sphere is recovered in
the long-time limit. For both navigation strategies, the relaxation times
towards the stationary distribution depend on the particular value of the Hurst
parameter. We also show that on $\mathbb{S}^{2}$, scaled Brownian motion,
distinguished by a time-dependent diffusion coefficient with a power-scaling,
is independent of the navigation strategy finding a good agreement between the
analytical calculations obtained from the solution of a time-dependent
diffusion equation on $\mathbb{S}^{2}$, and the numerical results obtained from
our numerical method to generate ensemble of trajectories.
