?

You can hide the Activity tab from guests in the ACP > System > Site Features > Menu Manager. Select "Activity" and change the "Who can see this item?" settings.

Next, we just need to set these 3 application settings. 

Go to ACP > System > Applications and select "System". There are a few areas you'll want to select the lock icon for: Content Discovery, Search, and Activity Stream. Uncheck "Guests" for these 3 areas and you should now have things set up the way you want.