In short, look for a pulsar orbiting a black hole, and see if the signal deviates from relativistic predictions.

From the article in more detail:

Some theories, like one proposed by Steven Giddings at the University of California, Santa Barbara, in 2014, predict that the black hole’s internal state can be linked to quantum fields outside, in the black hole’s “atmosphere”. This coupling would show up as fluctuations in the space-time around the black hole.

If a pulsar is orbiting it, its radio signal will look normal whenever the pulsar passes in front of the black hole. But when the black hole eclipses the pulsar, the radio beam will reach us via a region of space-time that is steeply curved by the immense gravity.

General relativity predicts that as a result, the signal will arrive early or late at our radio telescopes, with the discrepancy altering smoothly as the pulsar orbits. Quantum gravity, however, says the fluctuating space-time will alter the signal in irregular ways – such that a graph of the arrival times will look “fuzzy”.

Studying a fuzzy pulsar could confirm Giddings’s version of quantum gravity. Kavic and his colleagues propose searching for pulsar-black hole pairs using planned instruments such as the Square Kilometre Array and the Event Horizon Telescope (https://arxiv.org/abs/1607.00018v3).

The arXiv paper makes interesting reading too. First, the citation:

Shining Light On Quantum Gravity With Pulsar-Black Hole Binaries by John Estes, Michael Kavic, Matthew Lippert & John H. Simonetti, arXiv, 28th February 2017

Estes et al, 2017 wrote:Pulsars are some of the most accurate clocks found in nature, while black holes offer a unique arena for the study of quantum gravity. As such, pulsar-black hole (PSR-BH) binaries provide ideal astrophysical systems for detecting the effects of quantum gravity. With the success of aLIGO and the advent of instruments like the SKA and eLISA, the prospects for the discovery of such PSR-BH binaries are very promising. We argue that PSR-BH binaries can serve as ready-made testing grounds for proposed resolutions to the black hole information paradox. We propose using timing signals from a pulsar beam passing through the region near a black hole event horizon as a probe of quantum gravitational effects. In particular, we demonstrate that fluctuations of the geometry outside a black hole lead to an increase in the measured root mean square deviation of the arrival times of pulsar pulses traveling near the horizon. This allows for a clear observational test of the nonviolent nonlocality proposal for black hole information escape. For a series of pulses traversing the near-horizon region, this model predicts an rms in pulse arrival times of ∼30 μs for a 3M⊙ black hole, ∼0.3ms for a 30M⊙ black hole, and ∼40s for Sgr A*. The current precision of pulse time-of-arrival measurements is sufficient to discern these rms fluctuations. This work is intended to motivate observational searches for PSR-BH systems as a means of testing models of quantum gravity.

Further on in the paper, we have this:

Estes et al, 2017 wrote:It is currently an open question how quantum gravity resolves the black hole information paradox. Although most attempts to answer it have employed purely theoretical arguments, observational data may provide answers or at least constraints. While some alternatives, such as the firewall scenario (Almheiri et al. 2013), predict phenomena that are extremely difficult to detect, other recent proposals feature large-scale nonlocal effects, e.g., Giddings & Lippert (2004), Papadodimas & Raju (2013), Dvali & Gomez (2014), Freidel et al. (2015), Hawking et al. (2016). In particular, the nonviolent nonlocality proposal of Giddings (2012, 2013b, 2014a,b, 2016) has potentially observable consequences. Thus, the practical question is, where is the best place to hunt for these elusive signals of quantum gravity?

The Event Horizon Telescope (EHT) has generated substantial interest in the possibility of observing quantum gravitational effects in the near-horizon region of Sagittarius A* (Sgr A*), the supermassive black hole at the center of the Milky Way galaxy. Direct observations of the accretion disk might exhibit modification of structures, such as the black hole shadow and the photon ring, predicted by general relativity (Giddings 2014b).

Another suggested method to investigate the near-horizon region, inspired by recent advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) discoveries (Abbott et al. 2016b,a), is to observe the gravitational waves emitted by the merger of two black holes. One could hope to detect deviations in the signal from the general relativistic prediction (Giddings 2016). The potential of gravitational wave astronomy has justifiably generated a great deal of excitement, but with so far only a handful of observed events so far, it is at best a promising but untested technique. Furthermore, both the substantial observational noise and the theoretical uncertainties inherent in the difficult numerical modeling of the inspiral process limit the ability to discern the effects of quantum gravity.

Classically, the structure of spacetime is established by sending clock readings between observers on a spatial coordinate grid. To explore quantum mechanical effects on spacetime, we suggest using timing signals from the most precise natural clock—a pulsar—carried by light beams through the region near an event horizon. Therefore, we propose that a pulsar-black hole (PSR–BH) binary is an ideal astrophysical system for observing quantum gravitational effects. The fact that such a system is in some sense “clean,” that is, not muddled by other astrophysical features, allowed for precision measurements to be taken of a novel gravitational phenomenon, gravitational waves (Taylor & Weisberg 1982; Weisberg et al. 2010; Weisberg & Huang 2016). It is this critical property of such binary systems that allows them to be used as effective probes of quantum gravity.

Pulsar–neutron star (PSR–NS) binaries are very clean systems whose orbital parameters can be measured with extreme accuracy, and for this reason, they have played a long and valuable role as precision astrophysical laboratories. Most famously, observations of the so-called “binary pulsar” (PSR 1913+16) were the first to confirm the existence of gravitational waves (Taylor & Weisberg 1982). In addition, general relativistic effects, such as the Shapiro time delay of pulsar radiation passing through the gravitational potential well of the companion NS, have been measured, yielding precision tests of GR (Kramer et al. 2006).

Further on, the authors provide this:

Estes et al, 2017 wrote:One particularly interesting alternative scenario proposed by Giddings (2012, 2013b) suggests that modifications of local quantum field theory appear as long-wavelength fluctuations set by the Schwarzschild scale rather than the Planck scale. Note that for a macroscopic black hole of mass MBH, the Schwarzschild radius Rs = 2GMBH/c2, which completely characterizes classical, non-rotating black holes, is many orders of magnitude larger than the Planck length lp = [hbarG/c3]½. These are strong but low-energy fluctuations, with a large amplitude and a mild impact; infalling observers, for example, still travel unharmed through the horizon region. As a result, this proposal is termed “nonviolent nonlocality.”

Basically, if the black hole has sufficient mass, then the deviation of measured arrival times for the pulsar signal from the relativistic predicted value could be well within detectable limits. Given that astronomers have the ability to measure signal time or arrival to better than 1 part in 109, a value of 30 microseconds for a black hole of 3 solar masses is easily detectable, a 0.3 millisecond deviation for a black hole of 30 solar masses is even better, and if Sagittarius A can be pressed into service in the experiments, then the deviation from relativistic prediction is a whopping 40 seconds.

It will be interesting to see if this idea actually yields some results. If it does, we could have our first observational window onto a possible theory of quantum gravity.