A quarantine tank and a hospital tank are not really the same tank. A QT tank should be set up for a long term operation where the fish can get used to their new surroundings, foods, feeding schedules, etc over a period of time. A Hospital tank is for medicating and a filter is not necessary. An air stone is all you need as it will not effect most medications. Many meds will kill off the beneficial bacteria so there is no use in having them in a hospital tank. Also, you need to follow the instructions of the medicines you use as to whether to feed the medicated fish and when to do water changes as well as how much to change. A QT tank is more for just observation and if the need arises, the fish would need to be put into a hospital tank. The hospital tank need not be running when no fish are sick.
Since most meds are packaged in 10 gallon dosages, a 10 or 15 gal
tank would make a good hospital tank. Just be aware that a 10 gal
tank does not hold 10 gallons of water so a single dose of a med might be a little stronger for more sensitive fish. With a 15 gal
tank, you can measure out 10 gals of water so your dosing will be more accurate. Just put a magic marker spot at the 10 gal
line so you just fill to the line and medicate. The process takes just minutes and you're set.