## 4 Discussion and Conclusions
The time series of the price returns did not reveal any asymmetry between the cases, in which either BTC or ETH is lagged, for \(q=2\). However, the time series of the volume traded and the number of transactions showed a small effect of such an asymmetry: if BTC led, the cross-correlation was slightly stronger, which we interpreted as an effect of faster information transfer in the direction BTC\(\rightarrow\)ETH than in the opposite direction, which we related to the BTC dominance on the cryptocurrency market. Interestingly, for \(q=4\), this asymmetry is easy to notice in the case of the returns as well--a manifestation of a fact that the correlations associated with the returns of large amplitude are more direction sensitive than the correlations associated with the small and medium returns. Another observation was that, for short scales, the strength of the cross-correlations between the simultaneous time series representing BTC and ETH was the largest for the price returns, whereas for the volume and the number of transactions, it was substantially smaller. This effect was less pronounced for the lagged time series, and it gradually disappeared for the longer scales. A general conclusion that we can draw from this study is that the overall fractal properties of the major cryptocurrency time series are stable regarding the pre-COVID-19, COVID-19, and post-COVID-19 periods. We can interpret this stability as related to the fact that the processes that govern the fractal organisation of the cryptocurrency trading data are so fundamental that they can resist the social and economical forces perturbing the market. This conclusion apparently challenges some earlier reports based on stock market data, whose fractal properties were market-phase dependent (e.g., [60]), but actually it must be noticed that here, we did not attempt to decompose the market dynamics, and we presented the average market properties only. As the fractality indicates that the processes driving the cross-asset trading characteristics are largely scale invariant, our result can be of significance from the market-modelling perspective. It also provides further support for the thesis that the cryptocurrency market has reached maturity, at least from the statistical and dynamical points of view. It should be noted, however, that our study has its inherent limitations. We analysed two major cryptocurrencies only; it is thus conceivable that results for other cryptoassets, especially those with much worse liquidity, would potentially diverge from the results presented here. We also stress that we restricted our analysis to a few values of time lag. If another study considered different lags, especially the shorter ones, both the the cross-correlation magnitude and the asymmetry effect could be different. We also analysed a particular time period only, while the history of the cryptocurrency market is much longer. We could thus expect that the properties of the lagged cross-correlations evolve in time as well. Each of these limitations should be overcome in future research based on extended data. Another issue to be considered in future is to focus on short periods of time in order to gain some insight into how the fractal cross-correlation magnitude and lag asymmetry depend on the market situation (the bull and bear markets, for instance). **Author Contributions:** Conceptualisation, S.D., J.K. and M.W. ; methodology, S.D., J.K. and M.W. ; software, M.W. ; validation, S.D., J.K. and M.W. ; formal analysis, M.W. ; investigation, S.D., J.K. and M.W. ; resources, M.W. ; data curation, M.W. ; writing--original draft preparation, J.K.; writing--review and editing, S.D., J.K. and M.W. ; visualisation, M.W. ; supervision, S.D. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Data Availability Statement:** Data available freely from Binance [34]. **Conflicts of Interest:** The authors declare no conflict of interest.