If you want to play QuizUp privately and keep your profile private you can turn on private Mode. To do so, please follow these steps:

When you've opened the QuizUp page (QuizUp tab), located in the middle of the tab bar you should see the Settings option located in the left corner on the top bar. 

On the Settings screen, turn Private Profile ON by tapping the toggle next to it (By tapping it again, you turn it off).

Now you have to approve other QuizUp users as friends before they can challenge you, chat with you or view your profile.

If you have any further concerns please submit a ticket.