I don't understand how this game could be a turn-based game. Do the players compete one against the other at the same time? (in this case, I don't understand where are the "turns" and what each player does). Do they play the same levels, one after the other? What happens when a player fails to pass a level? Do they only compete for the best score? (in this case, a simple leaderboard is enough).
The technical difficulty is not always higher for a real-time MP game. It depends strongly on what you want to do.