This TurboTax reference explains in detail the qualifications for filing Head of Household: https://turbotax.intuit.com/tax-tips/family/guide-to-filing-taxes-as-head-of-household/L4Nx6DYu9
Theoretically you could each file HOH if two different qualifying dependents were involved. But you cannot each use the same child as your "qualifying dependent."
No, you cannot both file as HOH unless each of you has a blood-related dependent to claim--and it cannot be the same child. The
custodial
parent can file a HOH and get the EIC and the childcare credit if you have a signed Form 8332; the other parent can claim the child and get the child tax credit.
Am I Head of Household?
https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household
https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household
If you qualify as Head of Household, when you enter your filing status (single or married filing separately) into Personal Info, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.
There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent** is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note, in particular, that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the exemption to him.
So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.
Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897
Scroll down to "Children of divorced or separated parents (or parents who live apart)"
** The custodial parent is the parent the child lived with for more than 183 days in 2018.
For tax purposes, there is no such thing as joint custody, regardless of what your legal agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent. The IRS goes by physical custody, not legal custody.