Hypothetically, if you were to win a tournament and the grand prize was something with a high Qi, would the person that wins have to have that same Qi? Or does it work differently?

Depends on if its a Staff based event. If its a user made event you'd have to have the Qi = to the item. If its a staff based event you don't need to have the Qi for it. For instance if the grand prize is an azurite force and you only have 500 qi you'd get it.

Be fore-warned though, if you do win a qi restricted item from a Nabi Sponsored event, you will more than likely not be able to trade the item with others unless they have the required qi.