Could you store your passwords in a phone app governed by a QR code?

Could ...? Sure. I think you meant Should ...? No, one shouldn't rely on QR code. Using QR code is similar to writing password on a sticker, but is even worse. The major difference is, that QR code is designed to be easily read by phones.

QR code can be duplicated. For instance, when you are waiting in line, many people around you can easily make picture of your QR code when you are using it, and then they can reuse whenever they want. Reading a text on your ID card from some distance can be hard. But QR code can be recognized much easier from the same distance.