What parts of QFT do not have classical analogs? We seem to be living in an era of classical analogs. A few examples:

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*There exist classical analogs to Hawking and Unruh radiation.


*As pointed out by Arnold Neumaier, in 1852, George Stokes "described all the modern quantum phenomena of a single qubit, explaining them in classical terms," including spin-1/2 systems.


*Lazaro and Link describe spin-1/2 systems in terms of a quaternionic mechanical system.


*Quantum tunneling can be easily described in a classical way.


*I have documented the classical analogs of quantum entanglement and the violation of Bell inequalities with Brownian motion, Ising models, classical optics and water waves.


*Considering stochastic QFT, SED provides a classical explanation for a number of different quantum effects.
Question: What remaining parts of QFT still escape any classical analog?
Edit: Scharf, following Einstein and Weinberg, gave a non-geometrical version of gravity that mainly acts as a classical field, although derived from a quantum gauge field, so GR is not my concern here.
Edit: An appeal to the Turing complexity of quantum computers is circular. Gil Kalai has made arguments against their realization in terms of the harmonic analysis of Boolean functions, but some would suggest it's underlying "analog" behavior that limits their realization. Spin is continuous, there's no "global phase" and amplitudes interfere, not probabilities. Furthermore, E-T uncertainty and Boltzmann's law put a strict lower bound on QC runtime. A cold computer is a slow computer. Landauer's limit ignores quantum fluctuations.
 A: In some sense, what a classical analog boils down to is that you can build an analog computer that solves some equation that shows up in a quantum mechanical problem. I find it hard to imagine that there are systems of equations for which it is fundamentally impossible to build an analog computer that represents those equations. However, there are cases where it will be very difficult for a classical computer to represent a quantum mechanical system. "Very difficult" typically means in terms of the size of the analog system or the time needed to run it.
I would expect highly entangled quantum states with many qubits to be difficult to simulate classically, since the size of the classical computation will grow exponentially with the number of qubits. So, even though to date no system like this has been built, a several-thousand-qubit quantum computer implementing Shor's algorithm with fault tolerance would be very difficult for a classical computer to simulate.
The general area of "things that a quantum computer can efficiently do that a classical computer cannot efficiently do" is known as "quantum supremacy." As a non-expert in this field, my general impression from reading about developments in this area is that current quantum computers (of order 50 qubits) are "marginally better" than classical computers on some toy problems -- but there are a lot of technical arguments about one is really using "the best" classical algorithm and whether the toy problems are relevant or not. So, arguably, with today's technology, we haven't been able to create a quantum system that can't be simulated classically, in an efficient way.
However, there are also naturally occurring quantum systems that are difficult to simulate classically, and for which a quantum computer with many qubits would probably do a much better job than a classical computer. For example, this would include many body problems in quantum chemistry.
