Difference between a qt and a hospital tank...
There need not be. What describes a hospital tank can be a qt, IF you're willing to do very frequent filter changes, and you can do it as one tank. Follow with me. Understand the trade-offs.

A 'cycled' qt relies on bacteria. You can go as far as sand, rock, etc...or just sponges with bacteria. You put the fish in, you watch fish for four weeks. If you have to treat, you then move the fish to a hospital tank. And if it's a sand-rock...