I know that if we require the diagonal entries of $R$ to be positive, then the decomposition is unique, but how to prove the following theorem?
If $A=Q_1R_1=Q_2R_2$ are two distinct QR decompositions of a full rank, square matrix $A$ then $$Q_2=Q_1D$$ $$R_2=DR_1$$ For some square diagonal $D$ with entries $\pm1$.
Given any QR factorization, you can force the diagonal elements of $ R $ to positive by multiplying from the right by a diagonal matrix of that form... I think you can probably take it from there.
[edit] Hmmm, maybe in this situation it is better to say that any QR factorization can be changed to make the diagonal elements of $ R $ positive by inserting a diagonal matrix with $ \pm 1 $ on the diagonal as follows: $ Q R = Q D \tilde R $.