I've searched through the SOFTWARE\Classes and SOFTWARE\Microsoft subkeys, but couldn't find anything related to "spartan" or "edge". Given that Edge is still very new, there really isn't much information about this yet.
As an example, this is how I gather the information about the Internet Explorer Version via the registry on a remote machine:
$ieVersion=[Microsoft.Win32.RegistryKey]::OpenRemoteBaseKey('LocalMachine', $args[0]).OpenSubKey('SOFTWARE\Microsoft\Internet Explorer').GetValue('SvcVersion')