How do I unlock the National Dex?

I'm at roughly 67% completion of the Alola Dex, with ~200 caught, ~290 seen. After scanning QR codes, I've caught a Swinub and Cyndaquil. However, neither were registered in the Pokédex, since neither are Pokémon in the Alola Dex. This brought up the following question:

When will I unlock the National Dex?

Do I need to have seen all the Pokémon in the Alola Dex? Do I need to complete some specific quest, like catching/defeating the Ultra Beasts?

There is no National Dex in Pokémon Sun and Moon, at least not in the traditional sense.

The functionality of the National Dex has now been rolled into Pokémon Bank. When you use Pokémon Bank with one of the compatible games, there will now be an option on the menu (accessed by pressing the Y button) for 'Pokédex'. When you open it, any Pokémon you've caught or seen will be registered. You can do this for all 6 Pokémon games compatible with Pokémon Bank.

Registering a Pokémon will show its Pokédex entry for all the games it appears in, even if you haven't played some of those games, but any forms and shiny sprites will only show up if you register a game that has encountered that version of the Pokémon.

In addition, if a Pokemon does not appear in the Generation 7 Regional Pokédex, it will not have an entry for Generation 7 in the Pokédex in Pokemon Bank. You will only be able to view the entries for Generation 6.