There is no bridge or tunnel. One was started back in the 1920s and parts of it still remain but it was never completed: https://en.wikipedia.org/wiki/Staten_Island_Tunnel
It’s kind of a political hot potato few want to touch these days. A tunnel feels like a no brainier but a lot of NIMBY types in Staten Island don’t want a fast connection to the rest of the city as it would likely change the area dramatically.