MFS/Single/HOH

Based on the details you provided it sounds like the one can file HOH and the other single.  Since they lived apart for more than 6 months, I believe that is one of the main qualifiers.   

See this link for more details:

Head of Household Divorce | H&R Block (hrblock.com)

The one with the kids can probably file HoH.

The other can definitely not file as single, at least based on this explanation of the law in New Jersey ("where we are too dumb to pump our own gas!")

Did you know that there is no "Legal Separation" in New Jersey?

https://www.kaplandivorce.com/blog/legal-separation-in-new-jersey

Quickfinder and taxbook should both address this with Flowcharts. Both excellent

Agree with @BobKamman . Only the parent who the kid lived with is eligible to file HOH. The other must file MFS. 
New Jersey does not recognize a separation agreement notarized or not. They must file MFS or MFJ the taxpayer would have to file MFS. The parent with the kids in residence could file HOH

They would be better off filing MFJ.

At @BobKamman  seriously? I’ll be back in a little bit to tell you about New Jersey fuel.

do they have a document from the court? Notarized agreement is between the parties. Research the tax law addressing whether they qualify for MFS or single. Then you can determine whether either qualifies for HOH. I believe they are MFS.