If you live together unmarried with the other parent, then no. Only one parent can claim the child and that parent gets all the tax benefits. The other parent should not even list the child on their return.
If you are divorced or separated, and share custody, and you had custody less than half the nights of the year, you can't claim the childcare credit.
If you are divorced or separated, and share custody, and you had custody more than half the nights of the year, then only you can claim the childcare credit. The other parent can't claim anything unless you give them a signed release form, and they can only claim the child tax credit. The ability to claim the child care credit or qualify for head of household always stay with the parent who had custody more than half the nights and can't be shared, transferred or given away. In Turbotax, list the child and follow the interview questions carefully, make sure to select the child lived with you more than half the year or 7 months or more (since the program thinks that 6 months means exactly half the year and exactly half is not more than half).
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