at least for me.
My Firewall-Log doesn't show blocked connections and I also bypassed the firewall, so most likely it isn't that, but: how about telling us which DNS hosts are involved and which ports (I guess all is handled by TCP port 443 TLS connections, but who knows).
Also: How can I generate a logfile to debug such problems? I don't see one in the Galaxy installation folder and also none in AppData\local\GOG.com\Galaxy. Neither do I see an option in Galaxy itself and I'm too lazy to look up all strings in the exe to find possible commandline parameters.
So ... how do I solve the Problem?
Off-Topic: WHEN do we get a sorting by purchase date for GOG games again? I don't ask for other libraries, as I guess the APIs doesn't expose this information, but it was in Galaxy 1.x and we want it back!
PS: Most likely GOG will not answer this, it's somehow a standard by now, right? So thank you for helping to all other users in advance and shame on the GOG team.

What do you mean by broken ? Give us more details please, so the community can help you eventually.
Logs are located in %programdata%\GOG.com\Galaxy\logs\

Thank you very much! Never had the idea to check there.
I simply get an "Cloud Sync failed" error with every game. Will check the logs if I see something there.
Ok, the GalaxyClientService.log tells:

"Temporarily blacklisting process [procID]. Error Message: Failed to find the valid path for Process:[previous procID]. Error: 31"

Hm. The Stacktrace isn't very helpful to me.

The GalaxyClient.log tells:
"Cloud sync for product '[ProductID]' failed with reason: UndefinedFailure."

Well, I would have loved if it would be a defined failure. ;P