Putting sent private messages into a block
No need to use quicktabs. The tabs functionality is inherent if you use the privatemsg_filter addon module - inbox, sent messages and all mail are exposed as three tabs.

It was only needed to add a 4th tab with the option to send a new private message.

The tab already existed as a page, so adding it was simple using hook_menu_alter().

So the final code is: 

&lt;?php 
function MYMODULE_menu_alter(&amp;$items) {
  // Make the privatemsg new link a tab
  $items['messages/new']['type'] = MENU_LOCAL_TASK;
}
?&gt;