Stone was given three life sentences for the attacks
Michael Stone's conviction for the murders of Lin and Megan Russell was based on the evidence of "a career criminal", appeal court judges heard.
Stone, 44, was jailed in 2001 for killing the mother and her six-year-old child, and attempting to murder Josie Russell, in Chillenden, Kent, in 1996.
Stone allegedly confessed to Damien Daley, an inmate in Canterbury prison.
London's Court of Appeal heard there was no scientific or separate witness evidence linking Stone to the murders.
Edward Fitzgerald QC, defending Stone, said there was only circumstantial evidence.
"He (Daley) was a career criminal who admitted to lying to get by in life," said Mr Fitzgerald.
Daley, who was in a neighbouring cell while Stone was on remand, claimed that he heard a voice "through the pipework" confessing to the killings.
Mr Fitzgerald said: "Evidence of the self-confession category is recognised by the courts to be inherently unreliable.
He told the court the type of evidence given by Daley was "easily concocted and difficult to disprove".
He said Stone's case illustrated the dangers of reliance on prison informants and said it was a duty of the Crown to research and disclose the full history of the witness they were relying on.
"The fact that he was a heroin addict was concealed by Daley himself," he said.
"This evidence and evidence of psychological problems was not available at trial."
Stone, who was given three life sentences, always denied the attacks.
Dr Russell, 45, and her daughters were attacked by a man wielding a hammer as they walked home from a swimming gala.
Josie, who now lives with her father in North Wales, survived the attack, but suffered severe head injuries and brain damage.
She was nine at the time of the attack.
Stone was first convicted in 1998, but appeal judges quashed that conviction and ordered a retrial in 2001.
That year, Stone was convicted for a second time after the retrial in which Daley gave evidence.
Then in 2004, lawyers argued that Stone had not received a fair trial and won the right to a second appeal.
The case was adjourned and is expected to end on Tuesday or Wednesday.