Sugar spikes increase growth hormone activity and cause oxidative stress, which both contribute to acne. I doubt sugar spikes alone are sufficient to cause acne.

To over simplify. Yes sugar equals acne. It also equals bad health. Glucose floating through the bloodstream is very damaging and the true cause of heart disease, not cholesterol. So the body produces insulin to get it out of the bloodstream and into the cells. When you consume more glucose than your cells can take in, it keeps trying with the insulin but if the cells are full they aren't going to take in more. IF we hadn't invented added sugar and refined grain junk and sedetary work, this would rarely happen.

Eventually your cells will become resistant and your pancreas exhausted and you will get diabetes, like about half our population these days.

In the meantime, both the sugar and the insulin are inflamatory (and acne is an inflammatory condition as are many other health conditions). And the insulin stimulates other hormones like HGH and sex hormones involved in acne.