As of right now functionality for staying logged in doesn't work correctly. You often get logged out anyways. This will be fixed in the next couple of days. An option for how long you want to stay logged in will be available as well.
By default the stayed logged in option will keep you logged in for 6 months.
Note 2: This option (at first) will only be available for one IP. That means your home wifi and mobile data plan are two different IPs.
(Note that the stay logged in option should not be used on public devices.)