In order to assess whether quantum resources can provide an advantage over classical computation, it is necessary to characterize and benchmark the nonclassical properties of quantum algorithms in a practical manner. In this paper, we show that using measurements in no more than three out of the possible 3N bases, one can not only reconstruct the single-qubit reduced density matrices and measure the ability to create coherent superpositions, but also possibly verify entanglement across all N qubits participating in the algorithm. We introduce a family of generalized Bell-type observables for which we establish an upper bound to the expectation values in fully separable states by proving a generalization of the Cauchy-Schwarz inequality, which may serve of independent interest. We demonstrate that a subset of such observables can serve as entanglement witnesses for states of the quantum-approximate-optimization algorithm for the MaxCut problem (QAOA MaxCut), and further argue that they are especially well tailored for this purpose by defining and computing an entanglement potency metric on witnesses. A subset of these observables also certifies, in a weaker sense, the entanglement in GHZ states, which share the Z2 symmetry of QAOA MaxCut. The construction of such witnesses follows directly from the cost Hamiltonian to be optimized, and not through the standard technique of using the projector of the state being certified. It may thus provide insights to construct similar witnesses for other variational algorithms prevalent in the noisy intermediate-scale quantum era. We demonstrate our ideas with proof-of-concept experiments on the Rigetti Aspen-9 chip for ansatz containing up to 24 qubits.