From a technical perspective, you don't need to trust the government in these cases - provided that they implement the solutions with built-in privacy.
I just meant that there are technical approaches that don't necessarily involve centralized storage of everyone's movements and contacts, but achieve the same goals.
Singapore (and others, for that matter) has allegedly solved some of these issues in their soon-to-be open source contact tracing app [0].
They basically let every device keep track on itself and it's encounters - until a diagnosis is made.
Can't remember the details from there off the top of my head, but you'd either do a lookup via a central authority, or notify peer-to-peer, depending on what other mechanisms are in play (ephemeral/co-signed IDs, etc.)
The German's app will use bluetooth to track whom you're closed to on the device. When you get ill, the device notifies everyone that id 18911-2342-112312 has it and people who came in bluetooth range to you will get a notice to self isolate.