Three women and a man were treated for the effects of breathing in smoke following the fire in Murrayburn Green, Sighthill, on Sunday night.
It is thought abandoned furniture was deliberately set alight in the building's common close at about 21:00.
Two of the woman were taken to hospital for medical checks. A total of eight homes had been evacuated.
Police said they were keen to speak to anyone who could help them trace those responsible.
Insp David Ferguson of Corstorphine Police Station said: "Thankfully no-one was seriously injured in this incident but it could have been much worse.
"I am keen to hear from anyone who was in the area around 21:00 on Sunday night and who might have information that can assist us with our inquiries.
"I'd also like to remind residents not to leave items of furniture or rubbish in communal areas where they can be a target for fire-raising."
A blaze at a block of flats in an Edinburgh housing estate was started deliberately, police believe.