I know this post is old, but I found the solution to this issue for anyone else who is experiencing it!
I ran into this issue when I logged out of my Gmail and again when an update happened to my Galaxy S4 (SGH-I337).
The error message was "Unfortunately the processandroid.process.acore has stopped working."
Clear the app cache and data:
From the Home screen, press the Menu key.
Touch Settings.
Touch Applications.
Touch Manage Applications.
Touch All.
Select the application that is having issues.
Note: For the following errors:
android.process.acore: Select Contacts storage.
android.process.mms: Select Messages.
com.google.process.gapps: Select Download Manager or Downloads (At the bottom of the list). Make sure the Downloads Manager is enabled. If it isn't, enable it before proceeding.
TWlauncher process com.sec.android.app: Select TWLauncher.
Touch Clear data and Clear cache if they are available. This resets the app as if it was new, and may delete personal data stored in the app.
If you turned off sync, return to Accounts & sync and turn it back on. If necessary, import any contacts you exported to the SD card.
Uninstall all third-party party task managers, battery saver, or apps that close programs. For information, see Manage Applications: Manage memory on Android.
Restart the device.
Open the app to test if the problem happens again.
If the problem recurs, uninstall the app (if possible).
Wipe the cache partition to clear system memory.
Note: To find steps for your device, search Support Community for: wipe cache [device name]
For more details, I got the following information from this website:
Force close error or process stops unexpectedly... | T-Mobile Support
Hope this worked!