Answer
$$0$$
Work Step by Step
Stoke's Theorem can be defined as: $\iint_{S} curl F \cdot dS=\int_{C} F \cdot dr $
We will divide the sphere into upper and lower hemispheres, let us say $S_1, S_2$ , respectively.
This implies that $C$ is a circle in the $xy$ plane oriented in the counter-clockwise direction.
Therefore, $\iint_{S_1} curl F \cdot dS=\int_{C} F \cdot dr $
Now, we have: $\iint_{S_2} curl F \cdot dS=\int_{-C} F \cdot dr=-\int_{C} F \cdot dr $
This implies that
$$\iint_{S} curl F \cdot dS=\iint_{S_1} curl F \cdot dS+\iint_{S_2} curl F \cdot dS \\=\int_{C} F \cdot dr-\int_{C} F \cdot dr \\=0$$
