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The only parent with the automatic right to claim the child as a dependent is the parent where the child lives more than half the nights of the year. The custodial parent can release the dependent to the non-custodial parent by giving them a signed form 8332 dependent release. That allows the non-custodial parent to claim the child as a dependent and get the tax credit. But the ability to claim EIC, dependent care credit and head of household status always remains with the custodial parent and can't be waived, transferred or shared. (The IRS follows federal law which is superior to any state court order. There is no such thing as 50/50 custody, you have to count the number of night where the child lived and it can't be exactly equal in a normal year with an odd number of days.)
If the child lived with you more than half the nights of the year, you are automatically eligible for all the dependent benefits including the child tax credit and EIC/Dependent care. If the child is only eligible for EIC/Dependent care, that means you answered that you were going to release the child dependent claim to the other parent by signing form 8332.
If the child lived with you less than half the nights of the year, you have to indicate this by choosing "5 months" or less in the program. Turbotax will then ask if you have a signed form 8332 from the other parent. When you say yes, Turbotax will assign the child tax credit but not EIC or the dependent care credit. You must mail the original signed 8332 to the IRS within 3 days after e-filing.
If the other parent refuses to sign for 8332 for you, you can't claim them as a dependent. The IRS will not enforce your custody order. But you could go to family court and have the judge order the other parent to sign.
