Which ball do you want to have in front?
A noted you would need to Boolean subtract the elements which are supposed to be in front from those which are behind.
Depending on how you are cutting things out, it would also be necessary to Boolean union some outlines.
You may find the Trim Vectors tool useful as well.
Do you have the original graphic as an SVG which shows the layering clearly?
See:
for a bit on this sort of thing.