Zero is the elder sister of Lord Kirinmaru who informed her brother of the prophecy the Shikon Jewel had predicted, that he would meet his demise at the hands of beings that are neither human nor demon and an impossible entity surpassing time itself.

As such she wants Sesshoumaru and Rin's half-demon twin daughters, and Inuyasha and Kagome Higurashi's quarter-demon daughter dead to prevent the prophecy's fulfillment.

(Source: Inuyasha Wiki)