The　information-theoretic　definition　of　quantum　correlation,　e.g.,　quantum　discord,　is　measurement　dependent.　By　considering　the　more　general　quantum　measurements,　weak　measurements,　which　include　the　projective　measurement　as　a　limiting　case,　we　show　that　while　weak　measurements　can　enable　one　to　capture　more　quantumness　of　correlation　in　a　state,　it　can　also　induce　other　counterintuitive　quantum　effects.　Specifically,　we　show　that　the　general　measurements　with　different　strengths　can　impose　different　orderings　for　quantum　correlations　of　some　states.　It　can　also　modify　the　monogamous　character　for　certain　classes　of　states　as　well　which　may　diminish　the　usefulness　of　quantum　correlation　as　a　resource　in　some　protocols.　In　this　sense,　we　say　that　the　weak　measurements　play　a　dual　role　in　defining　quantum　correlation.