Attorney-General Mark Speakman on Thursday said the state would apply to the High Court for special leave to appeal the unsuccessful case in the NSW Court of Criminal Appeal.
The state wanted a 52-year-old man to face trial over the alleged murders of three Aboriginal children in the NSW town of Bowraville in the 1990s.
Evelyn Greenup, four years old, Clinton Speedy-Duroux, 16, and Colleen Walker, 16, disappeared from the northern NSW town over a five-month period from September 1990.

Clinton Speedy, who is believed to have been murdered near the Bowraville Aboriginal Mission in 1991. Source: Supplied
The man, who can't be named for legal reasons, was acquitted of Clinton's murder in 1994 and of Evelyn's murder in 2006.
If special leave is granted early in 2019, the attorney-general expects the High Court appeal itself to be heard in mid-2019.

Protesters want a man, who can't be named for legal reasons, to go to trial for murdering three children. Source: AAP