## 4 Discussion and Conclusions  
In this work, we studied the detrended cross-correlations between three trading characteristics of two major cryptocurrencies, BTC and ETH, over the last 2 years. This period was characterised by the profound stress imposed on the global economy by the COVID-19 pandemic, but after the initial shock of early 2020, when most of the markets experienced heavy losses, there was a quick rebound to new all-time highs in the second half of the year. During the pandemic, both the traditional markets and the cryptocurrency market passed through different stages, alternately governed by strong \"bears\" and \"bulls\". Recently, the military escalation in Ukraine has also been exerting a heavy impact on the markets, including the cryptocurrency market, leading them to suffer from extra drawdowns that added momentum to the bear market dominating the scene since December 2021. From the cryptocurrency perspective, another interesting structural phenomenon is an emergent, strong permanent coupling of the cryptocurrency market and the stock market [9; 44; 45; 46; 47]--a phenomenon that, prior to the pandemic, used to be observed only occasionally [8; 48; 49; 50; 51; 52; 53]. Financial market data are well-known for their nonstationarity. This is why the classical approach to quantifying correlations among such data in terms of, for instance, the Pearson coefficient can overlook important properties of the data. The formalism based on the detrended fluctuation analysis [54] and its further generalisations [37; 38; 39; 40] are much more suited for non-stationary signals, as their inherent feature is the elimination of multiscale trends. At present, this approach is favoured and becomes a standard tool of time series analysis. In our work, this formalism was employed to investigate cross  
Figure 7: **(Main panels)** Fluctuation functions \(F_{q}^{\rm XY}(\rm s)\) calculated for time series of the number of transactions \(N_{\Delta t}(t)\) in \(\Delta t\) for X = BTC and Y = ETH. In each panel, vertical dashed lines denote a range of time scales \(s\) for which a power-law model can be fitted to \(F_{q}^{\rm XY}(\rm s)\). Extreme values of \(q\) are also shown. Three cases are considered: both time series are simultaneous **(top)**, time series representing BTC is advanced by \(\tau=1\) min **(middle)**, and time series representing ETH is advanced by \(\tau=1\) min **(bottom)**. **(Insets)** The bivariate scaling exponent \(\lambda(q)\) vs. the mean univariate scaling exponent \(h_{\rm XY}(q)\) calculated for the same time series. Error bars denote the standard errors. correlations between two major cryptocurrencies--BTC and ETH. We selected three trading characteristics: price returns, volume, and number of transactions in a time unit (for which we chose 10 s). We did not restrict the analysis to the time series collated parallel in time, but we also analysed the ones that were shifted relative to each other by 1 min. By looking at the univariate and bivariate fluctuation functions, we found that both were manifesting the multifractal property. Only the results for the price returns can be directly compared with the analogous results obtained for the data covering earlier periods before COVID-19 [44; 55; 56; 57; 58; 59]. In this case, we see that the fractal properties of the price returns have not changed much. This is true not only if we look at the fluctuation functions, but also at the bivariate and mean univariate scaling exponents, which are almost equal in this case. We also analysed the lagged cross-correlations between the time series representing BTC and ETH for the first time in the literature. We found that shifting one of the time series suppresses the cross-correlation magnitude, but nevertheless, the remaining cross-correlations are still significant, especially for the scales larger than a few hours, during which the information that arrives on the market is able to disperse.