State tax filing

Yes and both.  According to tax law ONLY the custodial parent can claim the child unless they relinquish the dependent to the non-custodial parent with a signed 8332 form, however, if the divorce decree specified that the non-custodial parent can claim the dependent on specific years then the custodial parent can be held in contempt of court if they fail to issue the 8332 form for those years.

These are a paraphrase of the IRS rules for divorced or separated parents that live apart.

[Note: Unless the parents have been separated at all times during the last 6 months of the year, these rules do not apply.]

See “Children of divorced or separated parents or parents who live apart” in IRS Pub 501 for full information.

https://www.irs.gov/publications/p501#en_US_2018_publink1000220904

This assumes that the child is under age 18 (in most states).  Once the child becomes an adult (Emancipated child), custody becomes mute and these rules no longer apply.(See examples 5 & 6 in Pub 501 for more information)

There is no such thing in the Federal tax law as 50/50, split, or joint custody. The IRS only recognizes physical custody (which parent the child lived with the greater part, but over half, of the tax year. That parent is the custodial parent; the other parent is the noncustodial parent.)

Who can claim the exemption and credits depends on who is the custodial parent. (By the IRS definition of custodial parent for tax purposes - this is not the same as the legal custody that a court might grant.).

The test that the IRS uses to determine the custodial parent is where the child lived for more than 1/2 (or greater part) of the year. The IRS will go so far as to require counting the nights spend in each household - that person is the custodial parent for tax purposes (if exactly equal and more than 183 days - The custodial parent is the parent with the highest AGI, if less than 183 days then neither parent has custody so the child cannot be claimed by either parent). And yes they are that picky.

See Custodial parent and noncustodial parent  in Pub 501

https://www.irs.gov/publications/p501#en_US_2018_publink1000220906

Only the Custodial parent can claim: (Child would be listed as non-dependent EIC & CC only)
-Head of Household
-The Earned Income Credit
-The Child and Dependent Care Credit
-The Health Coverage Tax Credit

The non custodial parent can only claim: (Child would be listed as dependent)
- The child as a dependent
- The Child Tax Credit or credit for other dependents

But only if specifically specified in a pre-2009 divorce decree, separation agreement or the custodial spouse releases the exemption with a signed 8332 form - after 2009 the IRS only accepts a signed 8332 form that must be attached to the non-custodial parents tax return.

Note. If you are the non-custodial parent filing your return electronically, you must file Form 8332 with Form 8453, (U.S. Individual Income Tax Transmittal) for an IRS e-file Return. See Form 8453 and its instructions for more details. This must be done within 3 days of your e-filed return being accepted by the IRS.

This does NOT mean that the custodial parent can ignore any Decree or court order allowing the non-custodial parent to claim the exemption - they can be required to issue the 8332 form. They could be required by the court to do so or be in contempt.