"We carried out the attacks," Taliban spokesman Ehsanullah Ehsan told the AFP news agency by phone.
He said the two bombs, which wounded 82 people, were to avenge the arrest in Quetta of three people, including Younis al-Mauritani, believed to be a senior Al-Qaeda leader who had planned attacks abroad.
"We will launch a bigger attack in future," Ehsan said.
Wednesday’s attacks targeted Pakistan's paramilitary Frontier Corps, which was responsible for the capture of the Al-Qaeda operatives, police said.
One attacker detonated his bomb-laden car outside the residence of the Quetta deputy chief of the Frontier Corps, Farrukh Shahzad, and a second blew himself up inside the house, said senior police official Hamid Shakil.
Shahzad was wounded, his wife was killed and at least one of his children injured, security officials said.