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As a heavy user of the Twitter service a good client app is essential for me. Tweetings has become my go-to Twitter app due to its smooth operation and the fact it is compatible with multi-view on the Note 2.
It makes good use of swiping between columns and can be configured to work as desired.
Google Play Music
I listen to music all the time on the Note 2 and the Play Music app is very nice. The big display of the Note 2 makes it a breeze to search through my large music collection which is stored in the Google cloud.
The Google + social network is not for everyone, but those use use it will find the Android app to be best in class. It displays the news feed in an attractive layout the has updates sliding in and out as you swipe down the screen.
It can handle all functions of Google +, including video hangouts which are especially nice on the large screen of the Note 2.