When you are told that cookies are not saved, refresh the page (with Chrome press F12 (open Developer Tools https://developer.chrome.com/devtools/docs/shortcuts) and then hold the mouse down on the refresh icon and pick "Hard Reload" from the menu, to make absolutely sure the page is reloaded).
Maybe you are logged in, but your pages are served by a stale cache.
As your Safari works, that is equally the proof that AMASS works as designed. This site does not make anything differently based on browser you use.
In Chrome it is also easy to check that cookies are saved:
Open developer Tools
Click "resources" tab
If there is a cookie there with a long garbled name, size about 300 bytes, then that is the login cookie.
(just to make sure: please do NOT post a screenshot of that window)
See also https://support.mozilla.org/en-US/questions/1034392
I did what you suggested. There is no cookie from POTN at all in Chrome. I did the hard reload also. I tried 3 more times, all with the same result as before. As it is a completely fresh install there were only cookies for the 6 sites I logged into that I know needed cookies. The only one that did not work POTN. I checked Firefox as well, no cookies from POTN there either. That is also a completely fresh install of Firefox 34.0.5 with no add ons or extensions put on yet.
I think I may have been unclear with you. When I say Safari works, I mean that its the only one that doesn't log me out for no reason. If I close the Safari page, I will still be automatically logged out. For that reason I leave Safari open all the time. If for some reason I need to reboot the computer I have to go through all the steps to login and get my pages the way I liked them again.