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The $SL(2,\mathbb Z)$-symmetry of Cherednik's spherical double affine Heckealgebras in Macdonald theory includes a distinguished generator which acts as adiscrete time evolution of Macdonald operators, which can also be interpretedas a torus Dehn twist in type $A$. We prove for all twisted and untwistedaffine algebras of type $ABCD$ that the time-evolved $q$-difference Macdonaldoperators, in the $t\to\infty$ $q$-Whittaker limit, form a representation ofthe associated discrete integrable quantum Q-systems, which are obtained, inall but one case, via the canonical quantization of suitable cluster algebras.The proof relies strongly on the duality property of Macdonald and Koornwinderpolynomials, which allows, in the $q$-Whittaker limit, for a unifieddescription of the quantum Q-system variables and the conserved quantities aslimits of the time-evolved Macdonald operators and the Pieri operators,respectively. The latter are identified with relativistic $q$-difference TodaHamiltonians. A crucial ingredient in the proof is the use of the "Fouriertransformed" picture, in which we compute time-translation operators and provethat they commute with the Pieri operators or Hamiltonians. We also discuss theuniversal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations,for which we prove a duality relation, which simplifies the proofs further.

Macdonald Duality and the proof of the Quantum Q-system conjecture