Moving an object on the surface of a sphere while the sphere rotates

I have a sphere of size 1 that constantly rotates about the world’s x-axis. I spawn a cube at 0,0,1 and make it a child of the sphere so it rotates with the sphere. I’d like to have a function that will move the cube a small amount to its left or right while also reorienting itself so it lies flat on the sphere’s surface

I’m very confused as to the vector or quaternion math required to accomplish this so hopefully someone can help in that direction