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Certain polynomials in 
${{n}^{2}}$ variables that serve as generating functions for symmetric group characters are sometimes called 
$\left( {{S}_{n}} \right)$ character immanants. We point out a close connection between the identities of Littlewood–Merris–Watkins and Goulden–Jackson, which relate 
${{S}_{n}}$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra 
${{H}_{n}}\left( q \right)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood–Merris–Watkins identities and selected Goulden–Jackson identities that relate ${{H}_{n}}\left( q \right)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis–Lê–Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.
