How to use Copy Rotation constraint to copy Quaternion rotation?

There is no Quaternion in Copy Rotation’s Order option:

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What happens when the owner of the constraint has quaternion rotation mode? What if the target has quaternion rotation mode too? How can I only copy an axis of quaternion rotation?

Copy Rotation decomposes the quaternion rotation to euler rotations to end up with a matrix they use to figure out the new positions. Here is a discussion of the whole issue which will probably not be resolved because of the massive amount of coding required for little benefit gain.

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You could use a driver to copy the quaternion rotation.

Thank you very much. Now I have much better understanding of how constraints work.

You can write your own driver function that copies those values and then later call that function in the driver.