My scope is set to : https://www.googleapis.com/auth/plus.login
I know I am using the scope because when I change it, and for example add the email scope, the permissions requested change, and if I add an invalid scope, I get an error saying that my other scope(s) including the above mentioned are valid. So I am for sure using this scope.
On the G+ oauth API reference it says this scope should allow me to access Moments.
These are the permissions requested: http://i.imgur.com/7c0X1HZ.png
My oauth app/client on google developers dash is using the google plus API. I am able to get a token, and name/email when using certain scopes. This is for a web server application.
I also turned off the security settings for "trusted applications" on my test account.