So there are lots of problems with this including but solving the social problem of people not trusting systems and the fact that as presented at face value wouldn't provide any useful security if it worked. But it doesn't work because the contact tracing protocol is based on a common but fundamental misunderstanding of entanglement. Measuring one member of an entangled pair of qubits will always result in 50-50 regardless of if or how the partner has been measured. Only by reconnecting and comparing the results to see correlation can you see any effect of entanglement.
More generally quantum mechanics (including entanglement) doesn't provide any mechanism for remote communication of any sort. If anything you did locally affected the results of a remote collaborator even allowing for pre shared entanglement that could be used for faster than light communication which is impossible within the framework of standard quantum's mechanics and therefore quantum computers.
