As of today, there are two ways to have your proof of vaccination for a Covid 19 vaccination on your smartphone: the already known Corona App and the new CovPass App from the Robert Koch Institute. Both allow you to scan the QR code you received when you were vaccinated. If you have already been vaccinated and have not received a QR code, this can be generated with the existing documents in a pharmacy.
So now you have the possibility to prove via smartphone that you are vaccinated and you don’t have to walk around with the yellow vaccination card anymore. The only difference between the apps is that CovPass really only stores the vaccination record. The Corona app also comes with the tracing function.
If you want to travel within Europe, you have the advantage that the proof in CovPass is valid throughout Europe. The EU has previously agreed on a standard and should be sufficient for travel the smartphone.
Here you can find the apps for iOS, Android and Huawai (in the App Store, the app is not yet in the search):