Er ist nach Hause gekommen

vs

Wir sind zum Restaurant gelaufen

why “nach” in the first one and “zum” in the second one?

@kitten I think it has to do with the "class" of location?

Like, you would use "nach Berlin" same as "nach Hause", which are sort of fixed / "named" locations.

As opposed to "zum Restaurant" (aka "zu dem", but no one uses the long form).

Not exactly sure, but I think it mostly depends on if you're referring to a general location, vs a specific thing (like a restaurant). Given it would be "zum Haus". I think the difference is sort of the same as English "going home" vs "going to a restaurant", but I can't explain why it works that way either (as in, when you use "to" vs when not).

@kitten zu is followed by proper names (or like, more generic nouns like restaurant) while nach is for larger geographic locations (like cities, countries, continents, etc) and specifically “hause”