TITLE: Quaternions multiplication order (to rotate & unrotate)
QUESTION [7 upvotes]: Say, I have parent object rotation quaternion Qp & child object rotation (local - relative to parent) quternion Qch. 
1) Which order should I multiply them to get child object world(total) rotation QW?
QW = Qp * Qch or QW = Qch * Qp ? And what geometric interpretation of this order - same or reverse order of rotation execution?
2) And one more question: If I already have result total rotation of child object QW (calculated in proper way (see #1), I also know Qp & want to calculate Qch. Which order should I multiply Qp.Inverse & QW ?
3) And last - if we have situation #2, but opposite: QW & Qch are known, & Qp we need to get, what order of QW & Qch.Inverse multiplication should we use?
Thanks a lot!

REPLY [5 votes]: 1): QW == Qp * Qch It means we apply Qch 1st, & Qp then. So order of rotations applying is always from right to left.
2): Qch == Qp.Inversed * QW So we apply QW 1st, then unrotate it by Qp back.
3): Qp == QW * Qch.Inversed So we apply inverse Qch rotation. Then total QW. It yield Qp. 
Note: #2 & #3 work so, provided that QW is obtained by #1 formula.
Ps: This my answer is just concise translation to 3D-engines implementation the @DavidK's answer. See it for more fundamental understanding of, what is happened here under the hood.