Do I need to beat all the quests to go to the next difficulty?

I got Diablo 3 recently and have played up to Act III. I was just invited to a game by my friend who is in Act IV. We beat a level in Act IV and now it seems it is unlocked for me to play and to progress further in.

I can see the drawbacks of skipping quests, since I will be underleveled. However, I have very good gear (Rare/Legendary) that I got for good deals on the AH and I am able to defeat them.

So, if I were to play through Act IV while skipping Act III, would the game move me on to the next difficulty? Or would I have to complete those quests I skipped in Act III?

Thanks for reading.

Yes, you can move on the next difficulty without beating all the quests. You just need to beat the final boss to unlock the next difficulty. Beware that certain difficulties have a level cap, for instance, you can't enter inferno unless you are level 60.

I did this trick with a buddy of mine. He got me past the last boss, so I could train on the higher level monsters in the next difficulty.