  In this paper, we study optimal quadrature errors, approximation numbers, and
sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm
H}^{\alpha,\beta}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere
$\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the
approximation numbers for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>0$,
which gives a clue to Open problem 3 as posed by Krieg and Vyb\'iral in
\cite{KV}. Second, for the optimal quadrature errors for ${\rm
H}^{\alpha,\beta}(\Bbb S^d)$, we use the "fooling" function technique to get
lower bounds in the case $\alpha>d/2$, and apply Hilbert space structure and
Vyb\'iral's theorem about Schur product theory to obtain lower bounds in the
case $\alpha=d/2,\,\beta>1/2$ of small smoothness, which confirms the
conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open
problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators
and the least squares quadrature rules to obtain approximation theorems and
quadrature errors for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>d/2$ or
$\alpha=d/2,\,\beta>1/2$, which are order optimal.
