I don't think polygamy is specifically disallowed by the state, depending on how you define polygamy. Since marriage comes with tax and legal benefits, the state has an active interest in controlling what is legally defined as marriage (specifically, how many people are party to one marriage. I'm not talking about controlling any kind of two-party marriages like one man and one woman, two men, or two women).

For the religious definition of marriage, there are no laws saying you can't be married in your religion to multiple people. The only stipulation is that you can only be legally married to one person because of the aforementioned legal and tax benefits. There's no law prohibiting me from having six female roommates who are exclusively in a relationship with me. There is a law saying I can only be legally married to one of them at a time. The rest of the women would be legally single and religiously married.