Triply efficient shadow tomography aims to learn all expectation values of a set of observables from a quantum state with sample- and time-efficient protocols that use only a constant number of copies of the state at a time. This paper presents triply efficient shadow tomography protocols for three important sets of observables: k-local Pauli operators, k-body fermionic operators, and all n-qubit Pauli operators. The protocols use two-copy measurements and are based on fractional colorings of the commutation graph of the observables. For k-local Pauli operators, the protocols are already known to be triply efficient. For k-body fermionic operators and all Pauli operators, the paper provides the first triply efficient protocols. The protocols use two-copy measurements, which are necessary for sample-efficient shadow tomography. The paper also shows that for k-body fermionic operators, sample-efficient shadow tomography with single-copy measurements is impossible. The protocols use techniques from graph theory, including chi-boundedness, to reduce the problem to a fractional coloring problem. The results demonstrate that two-copy measurements are necessary and sufficient for Pauli and fermionic shadow tomography. The paper also provides a compressed classical representation of an n-qubit quantum state that allows efficient extraction of the expected value of any Pauli observable. The results have implications for quantum simulations and quantum chemistry.Triply efficient shadow tomography aims to learn all expectation values of a set of observables from a quantum state with sample- and time-efficient protocols that use only a constant number of copies of the state at a time. This paper presents triply efficient shadow tomography protocols for three important sets of observables: k-local Pauli operators, k-body fermionic operators, and all n-qubit Pauli operators. The protocols use two-copy measurements and are based on fractional colorings of the commutation graph of the observables. For k-local Pauli operators, the protocols are already known to be triply efficient. For k-body fermionic operators and all Pauli operators, the paper provides the first triply efficient protocols. The protocols use two-copy measurements, which are necessary for sample-efficient shadow tomography. The paper also shows that for k-body fermionic operators, sample-efficient shadow tomography with single-copy measurements is impossible. The protocols use techniques from graph theory, including chi-boundedness, to reduce the problem to a fractional coloring problem. The results demonstrate that two-copy measurements are necessary and sufficient for Pauli and fermionic shadow tomography. The paper also provides a compressed classical representation of an n-qubit quantum state that allows efficient extraction of the expected value of any Pauli observable. The results have implications for quantum simulations and quantum chemistry.