When looking for a no-show sock that will stay up, there are three key things that you want them to have: a heel cup, which helps the sock form snugly against the silhouette of the heel without leaving any loose fabric on the outside a heel tab, which sits above the heel of the shoe and both provides extra padding to prevent rubbing and an arch lock, which is a band of compression around the arch of the foot that helps prevent chafing, slipping, and blistering.
While this is much less of an issue with tall socks, the closer to no-show they get, the bigger the chance they have of falling down, and causing you discomfort. The biggest issue with socks of any length is getting a pair that doesn't fall and bunch up inside your shoe on your run.